Commutative Power-Sum Exponentiation Theory

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mathematical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

12 (2012), 15

Keywords: Fundamental, mega-overmathematics, power tower, commutative hyperoperation, exponentiation, tetration, negative base power theory, number scale transformation, general power-exponential function hyperefficiency theory, commutative power-sum exponentiation theory, chaos theory, fractal theory.

Introduction

Numbers with very small and very large absolute values [Wikipedia Large_numbers] are extremely important for real world modeling. Moreover, their role exponentially increases because of computer science evolution which requires the so-called scientific number representation, as well as the storage and handling of such numbers to avoid the permanent danger of "computing overflow".

In classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]), exponentiation as raising numbers to powers by Michael Stifel [1544], as well as power functions y = xⁿ with constant exponents n and exponential functions y = a^x with constant bases a are widely used. Some power-exponential functions with variable bases and variable exponents such as

y = x^x

and also iterated (nested) exponentials (power towers)

a^b^c^... = a^{b^[c^(...)]}

with multiply (repeatedly) raising bases to powers so that power exponents are powers themselves are well-known, see also [Wikipedia Tetration]. Leonhard Euler [1777] introduced the notation

exp_a(x) = a^x = a^x ,

which can be combined with function iteration notation fⁿ(x) giving

exp_aⁿ(x) = a^a^...^a^x

(with a used n times on the right-hand side). He also showed that the infinite power tower

a^a^...

defined as the limit of

a^a^...^a

(with a used n times), converges for e^-e ≤ x ≤ e^1/e as n goes to infinity, which roughly gives the interval from 0.066 to 1.44. In particular, at a = 2^1/2 , this limit equals 2. Hans Maurer [1901] already used modern tetration notation

ⁿa = a^a^...^a (with a used n times on the right-hand side).

Donald Ervin Knuth [1976] introduced his up-arrow notation

a↑n = a^n = aⁿ ,

a↑↑n = a^a^...^a (with a used n times on the right-hand side),

a↑↑n(x) = exp_aⁿ(x) = a^a^...^a^x (with a used n times on the right-hand side)

interpreting super-powers and super-exponential functions via using m arrows in expression a↑^m n(x). John Horton Conway [1996] chained arrow notation

a→n→2 = a^a^...^a (with a used n times on the right-hand side)

provides similar generalization via increasing the number 2 and, more powerfully, by extending the chain.

Albert Arnold Bennett [1915] proposed commutative hyperoperations sequence defined by the recursion rule

F_n+1(a , b) = exp(F_n(ln(a), ln(b))

beginning with

F₀(a , b) = ln(e^a + e^b) = ln(e^a + e^b),

addition (I)

F₁(a , b) = a + b ,

multiplication (II)

F₂(a , b) = ab = e^{ln(a) + ln(b)} ,

a commutative form of exponentiation (III)

F₃(a , b) = e^{ln(a) ln(b)} ,

F₄(a , b) = e^{e^[ln(ln(a))ln(ln(b))]}

not to be confused with tetration [Wikipedia Hyperoperation].

Wilhelm Ackermann [1928] defined the function

φ(m , n , p)

resembling the hyperoperation sequence with reproducing such basic operations as addition, multiplication, and exponentiation at p = 0, 1, 2, respectively:

φ(m , n , 0) = m + n ,

φ(m , n , 1) = mn ,

φ(m , n , 2) = m^n = mⁿ ,

φ(m , n , p) = m↑^p-1 (n + 1)

for p > 2 with extending these basic operations using Knuth's up-arrow notation.

Reuben Louis Goodstein [1947] introduced the hyperoperations sequence of operations extending succession (the 0th) 1 + b , addition (the 1st) a + b , multiplication (the 2nd) ab , and exponentiation (the 3rd) a^b and gave the extended operations beyond exponentiation the Greek names tetration (the 4th)

a↑↑b ,

pentation (the 5th)

a↑↑↑b = a↑³ b ,

hexation (the 6th)

a↑↑↑↑b = a↑⁴ b ,

etc., where each operation is defined by iterating the previous one.

All this is used for numbers with so-called very small and very large absolute values [Wikipedia Large_numbers].

But common approaches have many disadvantages:

1) investigating already available possibilities is much less efficient than concertedly creating new possibilities;

2) positive number bases only are usually considered;

3) bases between 0 and 1 are not efficiently used for representing numbers with so-called very small and very large absolute values;

4) a uniform number scale is not suitable for creating hyperoperation hierarchy;

5) known number scale transformations such as using logarithmic scales cannot provide suitably simultaneously representing numbers both with very small and very large absolute values of the both signs;

6) natural numbers (positive integers) of multiple (combined, composite) power exponents only are usually considered;

7) multilevel placing multiple power exponents brings many typesetting difficulties and misunderstanding, especially by text transformation via software including browsers;

8) already usual exponentiation a^b is noncommutative and nonassociative, e.g.

2³ = 8 ≠ 9 = 3²,

2^3^4 = 2^(3^4) = 2⁸¹ ≠ 2¹² = (2^3)^4,

because in

a^b = e^{b ln(a)}

the roles of a and b are very different;

9) a commutative form of exponentiation (III)

F₃(a , b) = e^{ln(a) ln(b)} = a^ln(b)

by Albert Arnold Bennett [1915] provides noninteger values by natural a , b > 1 and growth much more slower than that of a^b by great a , b , which is a very important disadvantage when applying this commutative form of exponentiation to representing great numbers;

10) individual quantities of operands and operation results are not considered at all.

Therefore, in classical mathematics, both power functions and exponential functions providing often useful high orders of growth especially by multiply (repeatedly) raising bases to powers have very bounded domains of definition and efficiency.

Hence classical mathematics cannot (and does not want to) regard (adequately solve and even consider) very many typical urgent problems in science, engineering, and life. This generally holds when solving valuation, estimation, discrimination, control, and optimization problems, as well as in particular by measuring very inhomogeneous objects and rapidly changeable processes [Encyclopaedia of Physics 1973]. This is also very important for chaos theory (Ilya Prigogine [1993, 1997]) and fractal theory (Benoît Mandelbrot [1975, 1977, 1982]).

Mega-overmathematics by Lev Gelimson [1987-2012] based on its uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations provides universally and adequately modeling, expressing, measuring, evaluating, and estimating general objects. This all creates the basis for many further developing, extending, and applying mega-overmathematics fundamental sciences systems. Among them are, in particular, negative base power theory which defines raising a negative number to a power, general power-exponential function hyperefficiency theory which creates principally new possibilities providing number scale transformation, and present general exponentiation theory which generalizes exponentiation with creating its commutative replacements suitable for developing hyperoperation hierarchy.

Principal Ideas

Possible ideas are very natural:

to consider general power-exponential functions as unary operations;

to commutatively extend them to make them applicable to any quantities of operands;

to further generalize such functions via operand role individualization.

Sign-Conserving Multiplication

In classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]), usual multiplication is well-known. If all the factors to be multiplied are positive, then their product is positive, too, which is natural. If some factors to be multiplied are positive whereas the remaining factors to be multiplied are negative, then their product is positive when the number of negative factors is even whereas their product is negative when the number of negative factors is odd. This seems to be rather artificial than natural. Its origin (source) is that in the (real or complex) numbers, multiplication of real numbers distributes over addition in the well-known concepts of a ring and a field which both are commutative.

Nota bene: Both in mathematical logic and in set theory, the both distributive laws (of multiplication over addition and of addition over multiplication) hold. In mathematical logic, namely in Boolean algebra, logical disjunction ∨ plays the role of addition whereas logical conjunction ∧ plays the role of multiplication, and for any sentences (propositions that may be true or false) A , B , and C , we have both

(A ∨ B) ∧ C = (A ∧ C) ∨ (B ∧ C)

and

(A ∧ B) ∨ C = (A ∨ C) ∧ (B ∨ C).

In set theory, namely in set algebra, unification ∪ plays the role of addition whereas intersection ∩ plays the role of multiplication, for any sets A , B , and C , we have both

(A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)

and

(A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C).

In real or complex algebra, for any numbers A , B , and C , multiplication distributes over addition

(A + B)C = AC + BC

whereas addition does not distribute over multiplication because generally (as a rule)

AB + C ≠ (A + C)(B + C).

Therefore, already in the well-known concepts of a ring and a field which both are commutative, one of the the both distributive laws (of multiplication over addition and of addition over multiplication) does not hold. Hence it is possible and at least equally natural to additionally consider completely nondistributive rings and fields along with well-known rings and fields which are partially nondistributive.

Nota bene: Common multiplication naturally leads to noncommutative common power and exponential functions well-defined by negative bases if and only if exponents are integer whereas sign-conserving multiplication naturally leads to commutative sign-conserving power and exponential functions well-defined by any real-number bases and exponents.

The fundamental ideas of sign-conserving multiplication of real numbers are as follows:

the modulus (absolute value) of the sign-conserving product (as a result of sign-conserving multiplication) of real numbers equals the modulus (absolute value) of the usual product of these numbers;

the value of the sign function of the sign-conserving product of real numbers vanishes if and only if the value of the sign function of the usual product of these numbers vanishes, i.e. if and only if at least one of these numbers vanishes;

the value of the sign function of the sign-conserving product of real numbers equals 1 if and only if all these numbers are positive;

the value of the sign function of the sign-conserving product of real numbers equals -1 if and only if at least one of these numbers is negative and none of these numbers vanishes.

To denote sign-conserving multiplication, simply use the parenthesis " either instead of a multiplication sign if it is implicit (i.e. omitped) or to the left of a multiplication sign (e.g. × , • , Π , etc.) if it is explicitly used.

Analytically, for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their sign-conserving product

"Π_j∈J a_j = min(sign a_j | j ∈ J) |Π_j∈J a_j|

so that

|"Π_j∈J a_j| = |Π_j∈J a_j|

and

sign(Π_j∈J a_j|) = sign|Π_j∈J a_j| min(sign a_j | j ∈ J) = min(sign|a_j| | j ∈ J) min(sign a_j | j ∈ J).

Example: For any real numbers a , b , and c ,

a"b"c = a "× b "× c = a "• b "• c

= min(sign a , sign b , sign c) |abc|

so that

|a"b"c| = |abc|

and

sign(a"b"c) = sign|abc| min(sign a , sign b , sign c)

= min(sign|a|, sign|b|, sign|c|) min(sign a , sign b , sign c).

Nota bene: Introducing an additional factor, e.g.

sign|abc| = min(sign|a|, sign|b|, sign|c|),

is here necessary to provide

sign(a"b"c) = 0

if at least one of these numbers a , b , and c vanishes whereas then

a"b"c = 0

due to vanishing the factor |abc| which is absent in sign(a"b"c).

Tetration with Possibly Noninteger Multiplicity

To begin with, use power-exponential function definition

y = f(x) = ^ax = x^^a = x^² a = exp_x^[a]+1({a}) = x^x^...^x^{a}

with any positive a and x used [a] + 1 times on the right-hand side

where

a = [a] + {a}

is a positive (possibly noninteger) number,

[a] = floor(a) = entier(a) = max{z ∈ Z | z ≤ a} ≤ a

(Z is the set of all the integers)

is the integer part of a as the greatest integer not exceeding a , and

{a} = a - [a] ∈ [0, 1),

i.e.

0 ≤ {a} < 1,

is the fractional part of a as a sawtooth function.

In particular, by 0 ≤ a ≤ 1, we simply have

y = f(x) = ^ax = x^^a = exp_x^[a]+1({a}) = x^a = x^a ,

which behaves not better than y = x and is hence uninteresting.

To understand the naturalness of this sophisticated definition, consider the following example for

a = 1.5

with

[a] = 1,

{a} = 0.5:

^ax = ^1.5x = x^x^0.5.

This is natural because

¹x = x¹ = x = x^x^0,

²x = x^x = x^x = x^x^1

and the power exponent

x^0.5 = x^0.5

in

^1.5x = x^x^0.5

is the geometric mean value of the power exponents x^0 = 1 in

¹x = x¹ = x = x^x^0

and x^1 = x in

²x = x^x = x^x = x^x^1.

To further generalize this result, take a with the same [a] = 1 (which is here inessential because the only two last power exponents are relevant) and any {a} with

0 ≤ {a} < 1:

a = 1 + {a}

where

1 ≤ {a} < 2.

^1+{a}x = x^x^{a}.

This is natural because

¹x = x¹ = x = x^x^0,

²x = x^x = x^x = x^x^1

and the power exponent

x^{a}= x^{a}

in

^1+{a}x = x^x^{a}

is the weighted geometric mean value of the power exponents

x^0 = 1

with its natural weight 1 - {a} in

¹x = x¹ = x = x^x^0

and

x^1= x

with its natural weight {a} in

²x = x^x = x^x = x^x^1.

In fact,

[(x^0)^1-{a}(x^1)^{a}]^{1/[(1-{a})+{a}]} = [1^1-{a}x^{a}]¹ = x^{a} ,

quod erat demonstrandum.

Tetration Transformation Algorithm

To begin with, consider the well-known tetration notation

ⁿx = x^^n = x^x^...^x

with x used n times (n ∈ N = {1, 2, 3, ...}) on the right-hand side, namely always one (the first) time as a base and (if n > 1) further n - 1 times as exponents.

Namely, use the following algorithm:

1) separate the sign of argument (variable) x from its modulus (absolute value) |x|;

2) directly and explicitly assign sign x to the function value itself;

3) replace the argument (variable) x with its modulus (absolute value) |x|;

4) replace each exponent |x| with the maximum max(|x|, 1/|x|) of |x| and its inverse 1/|x|;

5) consider the product whose first factor

sign x = 0 (x = 0)

to vanish independently of the second factor, or, alternatively, which is sufficient, take its (zero) limit as its value.

General Power-Exponential Function Notation and Sense

Let us introduce:

a notation for powers and exponentials with single-level placing multiple power exponents via separating them with the backslash sign \ , e.g.

a^b^c^d = a^b\c\d ;

a space-saving notation for the sign function

a° = sign a ;

a space-saving notation for the function

a? = max(a , 1/a).

Also consider the well-known tetration notation

ⁿx = x^^n = x^x^...^x

(with x used n times on the right-hand side) by n = 2:

²x = x^^2 = x^x = x^x .

The above tetration transformation algorithm leads to the function

y = f(x) = x°|x|^|x|? = (sign x) |x|^{max(|x|, 1/|x|)} = (sign x) |x|^{(|x|+1/|x|+||x|-1/|x||)/2}

where by x° = sign x = 0 (x = 0), the second factor

|x|^|x|? = |x|^{max(|x|, 1/|x|)} = |x|^{(|x|+1/|x|+||x|-1/|x||)/2}

is not considered at all, or, alternatively, its (zero) limit is taken as its value.

We have piecewise y = f(x) =

x^x by x ∈ [1, +∞),

x^1/x by x ∈ (0, 1],

0 by x = 0,

-(-x)^1/(-x) by x ∈ [-1, 0),

-(-x)^(-x) by x ∈ (-∞, -1].

Now, beginning with the tetration notation ⁿx and using the parenthesis " (as well as in negative base power theory by Lev Gelimson [1987-2012]) between the base x and the number n of the base and the exponents each of which equals the base x , naturally introduce the simple notation for this function

y = f(x) = ²"x = x"^^2

and, more generally, for a function

y = f(x) = ⁿ"x = x"^^n

with giving it sense further.

Notata bene:

1. In negative base power theory, e.g. in

x"^x = x"^x = x°|x|^x = (sign x) |x|^x ,

the parenthesis " (placed to the right from the base x between the base x and the exponent which equals the base x in this case only) designates the following:

1.1) in the base x only, separate the sign of argument (variable) x from its modulus (absolute value) |x|;

1.2) directly and explicitly assign x° = sign x to the function value itself;

1.3) in the base x only, replace the argument (variable) x with its modulus (absolute value) |x|.

2. In power-exponential function hyperefficiency theory, e.g. in

ⁿ"x = x"^^n ,

the parenthesis " (placed to the left or to the right, respectively, from the base x between the base x and the optional operation signs with the number n of the base and the exponents each of which equals the base x) designates the following:

2.1) separate the sign of argument (variable) x from its modulus (absolute value) |x|;

2.2) directly and explicitly assign sign x to the function value itself;

2.3) replace the argument (variable) x with its modulus (absolute value) |x|;

2.4) replace each exponent |x| with the maximum

|x|? = max(|x|, 1/|x|)

of |x| and its inverse 1/|x|;

2.5) consider the product whose first factor

x° = sign x = 0 (x = 0)

to vanish independently of the second factor, or, alternatively, which is sufficient, take its (zero) limit as its value.

Power-Exponential Functions y = x"^^2 = x"^x = (sign x) |x|^max(|x|, 1/|x|), y = x^x , and y = x^(1/x)

Now systematically consider once more function

y = f(x) = ²"x = x"^^2.

To begin with, take the well-known power-exponential function

y = f(x) = x^x = ²x = x^^2 = x^x = e^{x ln x} .

Its first two derivatives are

y' = df(x)/dx = e^{x ln x} (ln x + x/x) = x^x (1 + ln x), f'(1) = 1;

y'' = d²f(x)/dx² = (x^x + x^x ln x)' = x^x (1 + ln x) + x^x (1 + ln x) ln x + x^x/x = x^x [(1 + ln x)² + 1/x], f''(1) = 2.

By x ∈ [1, +∞), this function behaves as desired (and much better than y = x^2n-1 and y = a^x) and has a suitable inverse.

By x ∈ [0, 1], the function y = f(x) = x^x brings nothing:

takes value 1 at x = 1;

has limit

lim_x→0+ f(x) = 1;

has the minimum

f(1/e) = (1/e)^(1/e).

To provide a suitable extension of this function to x ∈ [0, 1], it is logical to use

y = f(x) = x^1/x = e^{(ln x)/x}

with

y' = df(x)/dx = e^{(ln x)/x} [1/x² - (ln x)/x²] = x^1/x (1 - ln x)/x² , f'(1) = 1;

y'' = d²f(x)/dx² = [e^{(ln x)/x} (1 - ln x)/x²]' = e^{(ln x)/x} (1 - ln x)/x² (1 - ln x)/x² + e^{(ln x)/x} [(-2)/x³(1 - ln x) + 1/x²(-1/x)]

= x^1/x [(1 - ln x)² + 2x ln x - 3x]/x⁴ , f''(1) = -2.

Therefore, function y = f(x) = ²"x = x"^^2 = x"^x =

x^x by x ∈ [1, +∞),

x^1/x by x ∈ (0, 1],

0 by x = 0,

- (-x)^1/(-x) by x ∈ [-1, 0),

- (-x)^(-x) by x ∈ (-∞, -1]

compresses a uniform number scale between -1 and 1 and extends it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), see Fig. 1:

Fig. 1

Using the sign function and the maximum function provides unifying the above piecewise representations as

y = f(x) = ²"x = x"^^2 = x"^x

= x°|x|^|x|? = (sign x) |x|^max(|x|, 1/|x|) =(sign x) |x|^[(|x|+1/|x|+||x|-1/|x||)/2]

= (sign x) |x|^{max(|x|, 1/|x|)} = (sign x) |x|^{(|x|+1/|x|+||x|-1/|x||)/2}

where by

x° = sign x = 0 (x = 0),

the second factor

|x|^|x|? = |x|^max(|x|, 1/|x|) =|x|^[(|x|+1/|x|+||x|-1/|x||)/2] = |x|^{max(|x|, 1/|x|)} = |x|^{(|x|+1/|x|+||x|-1/|x||)/2}

is not considered at all, or, alternatively, its (zero) limit is taken as its value.

Examples:

²"3 = 3"^^2 = 3"^3 = ²3 = 3^^2 = 3^3 = 3³ = 27,

²"(1/3) = (1/3)"^^2 = (1/3)"^(1/3) = (1/3)^[1/(1/3)] = (1/3)^1/(1/3) = (1/3)³ = 1/3³ = 1/27,

²"0 = 0,

²"(-1/3) = (-1/3)"^^2 = (-1/3)"^(-1/3) = - (1/3)^[1/(1/3)] = - (1/3)^1/(1/3) = - (1/3)³ = - 1/3³ = -1/27,

²"(-3) = (-3)"^^2 = (-3)"^3 = - ²3 = - 3^^2 = - 3^3 = - 3³ = -27.

Notata bene:

1. The exponent is

1/|x| = |x|^-1 by 0 < |x| ≤ 1,

|x| = |x|¹ by 1 ≤ |x| < +∞

with the clear mirror symmetry (-1 and 1) of the exponent in the first additional level about |x| = 1.

2. The whole exponent |x|? = max(|x|, 1/|x|) of the base |x| has its minimum 1 by |x| = 1 whereas

lim_x→0 |x|? = lim_x→0 max(|x|, 1/|x|) = lim_x→±∞ |x|? = lim_x→±∞ max(|x|, 1/|x|) = +∞ ,

which provides much more efficiency than it is possible due to using both power and exponential functions.

3. This function y = f(x) is continuous together with its first derivative whereas the second derivative has discontinuity jumps at x = -1 and x = 1.

4. This function y = f(x) has three inflection points (with changes from being convex to concave or vice versa) at 0 (which is natural) and about ±0.582 where

y'' = d²f(x)/dx² = [e^{(ln x)/x} (1 - ln x)/x²]' = x^1/x [(1 - ln x)² + 2x ln x - 3x]/x⁴ = 0.

In particular, at x = 1,

(1 - ln x)² + 2x ln x - 3x = -2.

Power-Exponential Functions y = x"^^a = (sign x) |x|^max(|x|, 1/|x|)^^(a-1), y = x^^a = x^[x^^(a - 1)], and y = x^[(1/x)^^(a - 1)]

Let us investigate naturally generalizing the above functions

y = f(x) = x^x = ²x = x^x ,

y = f(x) = x^1/x ,

respectively, via iterating the exponents x and 1/x , respectively, any fixed (possibly noninteger) number of times.

To begin with, take power-exponential function

y = f(x) = ^ax = x^^a = exp_x^[a]+1({a}) = x^x^...^x^{a}

with x used [a] + 1 times on the right-hand side

where

a = [a] + {a}

is a positive (possibly noninteger) number,

[a] = floor(a) = entier(a) = max{z ∈ Z | z ≤ a} ≤ a

(Z is the set of all the integers)

is the integer part of a as the greatest integer not exceeding a , and

{a} = a - [a] ∈ [0, 1),

i.e.

0 ≤ {a} < 1,

is the fractional part of a as a sawtooth function.

In particular, by 0 ≤ a ≤ 1, we simply have

y = f(x) = ^ax = x^^a = exp_x^[a]+1({a}) = x^a = x^a ,

which behaves not better than y = x and is hence not interesting.

Further, e.g., by 1 < a ≤ 2, we simply have

y = f(x) = ^ax = x^^a = exp_x^[a]+1({a}) = x^x^(a-1) = exp[e^{(a-1) ln x} ln x].

Its first derivative is

y' = df(x)/dx = exp[e^{(a-1) ln x} ln x] [e^{(a-1) ln x} ln x]' = exp[e^{(a-1) ln x} ln x] e^{(a-1) ln x} /x [1 + (a - 1)ln x] = x^x^(a-1) x^a-2 [1 + (a - 1)ln x],

f'(1) = 1.

By x ∈ [1, +∞), this function behaves as desired (and much better than y = x^2n-1 and y = a^x) and has a suitable inverse.

By x ∈ [0, 1], this function brings nothing:

takes value 1 at x = 1;

has limit

lim_x→0+ f(x) = 1.

To provide a suitable extension of this function to x ∈ [0, 1], it is logical to use

y = f(x) = x^(1/x)^(a-1) = exp[e^{(1-a) ln x} ln x].

with

y' = df(x)/dx = exp[e^{(1-a) ln x} ln x] [e^{(1-a) ln x} ln x]' = exp[e^{(1-a) ln x} ln x] e^{(1-a) ln x} /x [1 - (a - 1)ln x] = x^(1/x)^(a-1) /x^a [1 - (a - 1)ln x],

f'(1) = 1.

Therefore, function y = f(x) = ^a"x = x"^^a =

x^x^(a-1) by x ∈ [1, +∞),

x^(1/x)^(a-1) by x ∈ (0, 1],

0 by x = 0,

-(-x)^[1/(-x)]^(a-1) by x ∈ [-1, 0),

-(-x)^(-x)^(a-1) by x ∈ (-∞, -1]

compresses a uniform number scale between -1 and 1 and extends it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), see Fig. 2:

Fig. 2. a = 2.9

Using the sign and maximum functions provides unifying the above piecewise representations as

y = f(x) = ^a"x = x"^^a = x°|x|^[|x|?^^(a-1)] = (sign x)|x|^max(|x|, 1/|x|)^(a-1) = (sign x)|x|^[(|x|+1/|x|+||x|-1/|x||)/2]^(a-1)

where by sign x = 0 (x = 0), the second factor

|x|^[|x|?^^(a-1)] = |x|^max(|x|, 1/|x|)^(a-1) = |x|^[(|x|+1/|x|+||x|-1/|x||)/2]^(a-1)

is not considered at all, or, alternatively, its (zero) limit is taken as its value.

Note that the last formulae hold by 1 < a ≤ 2 only.

Generally, by a > 1,

y = f(x) = ^ax = x^^a = exp_x^[a]+1({a}) = x^x^...^x^{a}

with x used [a] + 1 times on the right-hand side.

By x ∈ [1, +∞), this function behaves as desired (and much better than y = x^2n-1 and y = a^x).

By x ∈ [0, 1], this function brings nothing:

takes value 1 at x = 1;

has limit

lim_x→0+ f(x) = 1.

To provide a suitable extension of this function to x ∈ [0, 1], it is logical to use

y = f(x) = x^^(a-1)(1/x) = x^exp_1/x^[a]({a}).

Therefore, function y = f(x) = ^a"x = x"^^a =

^ax = exp_x^[a]+1({a}) by x ∈ [1, +∞),

x^^(a-1)(1/x) = x^exp_1/x^[a]({a}) by x ∈ (0, 1],

0 by x = 0,

- (-x)^^(a-1)[1/(-x)] = - (-x)^exp_1(-x)^[a]({a}) by x ∈ [-1, 0),

- ^a(-x) = - exp_-x^[a]+1({a}) by x ∈ (-∞, -1]

compesses a uniform number scale between -1 and 1 and extends it by (-∞ , -1] and [1, +∞).

Using the sign and maximum functions provides unifying the above piecewise representations as

y = f(x) = ^a"x = x"^^a = x°|x|^[|x|?^^(a-1)] = (sign x) |x|^[max(|x|, 1/|x|)^^(a-1)] = (sign x) |x|^{[(|x|+1/|x|+||x|-1/|x||)/2]^^(a-1)}

= (sign x) |x|^^(a-1)max(|x|, 1/|x|) = (sign x) |x|^^(a-1)[(|x|+1/|x|+||x|-1/|x||)/2]

= (sign x) |x|^exp_{max(|x|, 1/|x|)}^[a]({a}) = (sign x) |x|^exp_{(|x|+1/|x|+||x|-1/|x||)/2}^[a]({a})

where by sign x = 0 (x = 0), the second factor

|x|^[|x|?^^(a-1)] = |x|^[max(|x|, 1/|x|)^^(a-1)] = |x|^{[(|x|+1/|x|+||x|-1/|x||)/2]^^(a-1)} =

|x|^^(a-1)max(|x|, 1/|x|) = |x|^^(a-1)[(|x|+1/|x|+||x|-1/|x||)/2]

= x°|x|^exp_|x|?^[a]({x}) = |x|^exp_{max(|x|, 1/|x|)}^[a]({a}) = |x|^exp_{(|x|+1/|x|+||x|-1/|x||)/2}^[a]({a})

is not considered at all, or, alternatively, its (zero) limit is taken as its value.

Examples for a = 4:

⁴"3 = 3"^^4 = ⁴3 = 3^^4 = 3^3^3^3 = ⁴3 = 3^3²⁷ ,

⁴"(1/3) = (1/3)"^^4 = (1/3)^^(4-1)[1/(1/3)] = (1/3)^³3 = (1/3)^3^3^3 = (1/3)^3²⁷ = 1/(3^3²⁷),

⁴"0 = 0,

⁴"(-1/3) = - ⁴"(1/3) = - (1/3)"^^4 = - (1/3)^^(4-1)[1/(1/3)] = - (1/3)^³3 = - (1/3)^3^3^3 = - (1/3)^3²⁷ = - 1/(3^3²⁷),

⁴"(-3) = - 3"^^4 = - ⁴3 = - 3^^4 = - 3^3^3^3 = - ⁴3 = - 3^3²⁷ .

Notata bene:

1. The new exponent |x|? = max(|x|, 1/|x|) is

1/|x| = |x|^-1 by 0 < |x| ≤ 1,

|x| = |x|¹ by 1 ≤ |x| < +∞

with the clear mirror symmetry (-1 and 1) of the exponent in the first additional level about |x| = 1.

2. The new exponent |x|? = max(|x|, 1/|x|) of the base |x| has its minimum 1 by |x| = 1 whereas

lim_x→0 |x|? = lim_x→0 max(|x|, 1/|x|) = lim_x→±∞ |x|? = lim_x→±∞ max(|x|, 1/|x|) = +∞ ,

which provides much more efficiency than it is possible due to using both power and exponential functions.

3. This function y = f(x) is continuous together with its first derivative.

Exponentiation Commutativization

Let us investigate some natural approaches to making power-exponential functions commutative.

To begin with, consider commonly raising a real-number base b to a power with a real-number exponent r giving b^r . It is well-known that this operation is noncommutative because as a rule

b^r ≠ r^b .

Example: By b = 2 and r = 3, we have

2³ = 8 ≠ 9 = 3².

Commutative Power-Sum Exponentiation Type

Commutative Power-Sum Exponentiation Mode

Commutative Power-Sum Exponentiation

Idea

The principal idea of commutative power-sum exponentiation is as follows:

consider any set of real numbers;

if there are coinciding (equal) numbers among them, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) numbers to artificially differentiate between them;

consider all the permutations of these already distinguishable numbers;

for every permutation of these already distinguishable numbers, compose their power tower using them in the order corresponding to this permutation;

for all these permutations together, compose the sum of all these power towers;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative power-sum exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative power-sum exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative power-sum exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative power-sum exponential is denoted by

... ^⁺ a ^⁺ ... ^⁺ b ^⁺ ... ^⁺ c ^⁺ ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative power-sum exponential is denoted by

ΕΣ_j∈J a_j = ^⁺_j∈J a_j = ΕΣ{a_j | j ∈ J} = ^⁺{a_j | j ∈ J} = ... ^⁺ a_j' ^⁺ ... ^⁺ a_j'' ^⁺ ... ^⁺ a_j''' ^⁺ ...

for any n ∈ N = {1, 2, 3, ...}, a commutative power-sum tetration is denoted by

^⁺ⁿa = a ^⁺ a ^⁺ ... ^⁺ a

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a ^⁺ b = a^b + b^a = a^b + b^a ;

a ^⁺ b ^⁺ c = a^b^c + a^c^b + b^a^c + b^c^a + c^a^b + c^b^a ;

ΕΣ_{j∈{1, 2, 3}} a_j = ^⁺_{j∈{1, 2, 3}} a_j = ΕΣ{a_j | j ∈ {1, 2, 3}} = ^⁺{a_j | j ∈ {1, 2, 3}} = a₁ ^⁺ a₂ ^⁺ a₃

= a₁^a₂^a₃ + a₁^a₃^a₂ + a₂^a₁^a₃ + a₂^a₃^a₁ + a₃^a₁^a₂ + a₃^a₂^a₁ ;

2 ^⁺ 3 = 2^3 + 3^2 = 2³ + 3² = 17;

1 ^⁺ 2 ^⁺ 3 = 1^2^3 + 1^3^2 + 2^1^3 + 2^3^1 + 3^1^2 + 3^2^1

= 1^8 + 1^9 + 2^1 + 2^3 + 3^1 + 3^2 = 1⁸ + 1⁹ + 2¹ + 2³ + 3¹ + 3² = 24;

^⁺²a = a ^⁺ a = a^a + a^a = a^a + a^a = 2a^a ;

^⁺³a = a ^⁺ a ^⁺ a = = a₁ ^⁺ a₂ ^⁺ a₃

= a₁^a₂^a₃ + a₁^a₃^a₂ + a₂^a₁^a₃ + a₂^a₃^a₁ + a₃^a₁^a₂ + a₃^a₂^a₁

= a^a^a + a^a^a + a^a^a + a^a^a + a^a^a + a^a^a = 6a^a^a = 6 ³a ;

^⁺²2 = 2 ^⁺ 2 = 2₁ ^⁺ 2₂ = 2₁^2₂ + 2₂^2₁ = 2^2 + 2^2 = 2² + 2² = 8;

^⁺³2 = 2 ^⁺ 2 ^⁺ 2 = 2₁ ^⁺ 2₂ ^⁺ 2₃

= 2₁^2₂^2₃ + 2₁^2₃^2₂ + 2₂^2₁^2₃ + 2₂^2₃^2₁ + 2₃^2₁^2₂ + 2₃^2₂^21

= 2^2^2 + 2^2^2 + 2^2^2 + 2^2^2 + 2^2^2 + 2^2^2

= 2^4 + 2^4 + 2^4 + 2^4 + 2^4 + 2^4 = 2⁴ + 2⁴ + 2⁴ + 2⁴ + 2⁴ + 2⁴ = 96.

Commutative Power-Modulus-Sum Exponentiation

Idea

The principal idea of commutative power-modulus-sum exponentiation is as follows:

consider any set of real numbers;

if there are coinciding (equal) numbers among them, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) numbers to artificially differentiate between them;

consider all the permutations of these already distinguishable numbers;

for every permutation of these numbers, compose their power tower using them in the order corresponding to this permutation;

for all these permutations together, compose the sum of the moduli (absolute values) of all these power towers;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative power-modulus-sum exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative power-modulus-sum exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative power-modulus-sum exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative power-modulus-sum exponential is denoted by

... ^|^^|+ a ^|^^|+ ... ^|^^|+ b ^|^^|+ ... ^|^^|+ c ^|^^|+ ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative power-modulus-sum exponential is denoted by

|Ε|Σ_j∈J a_j = ^|^^|+_j∈J a_j = ^|^^|+{a_j | j ∈ J} =

= ... ^|^^|+ a_j' ^|^^|+ ... ^|^^|+ a_j'' ^|^^|+ ... ^|^^|+ a_j''' ^|^^|+ ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative power-modulus-sum tetration is denoted by

^|^^|+na = a ^|^^|+ a ^|^^|+ ... ^|^^|+ a

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a ^|^^|+ b = |a^b| + |b^a| = |a^b| + |b^a|;

a ^|^^|+ b ^|^^|+ c = |a^b^c| + |a^c^b| + |b^a^c| + |b^c^a| + |c^a^b| + |c^b^a|;

|Ε|Σ_{j∈{1, 2, 3}} a_j = ^|^^|+_{j∈{1, 2, 3}} a_j = ^|^^|+{a_j | j ∈ {1, 2, 3}} = a₁ ^|^^|+ a₂ ^|^^|+ a₃

= |a₁^a₂^a₃| + |a₁^a₃^a₂| + |a₂^a₁^a₃| + |a₂^a₃^a₁| + |a₃^a₁^a₂| + |a₃^a₂^a₁|;

(-2) ^|^^|+ (-3) = |(-2)^(-3)| + |(-3)^(-2)| = |-1/2^3| + |-1/3^2| = 1/2³ + 1/3² = 17/72;

(-1) ^|^^|+ (-2) ^|^^|+ (-3) = |(-1)^(-2)^(-3)| + |(-1)^(-3)^(-2)| + |(-2)^(-1)^(-3)| + |(-2)^(-3)^(-1)| + |(-3)^(-1)^(-2)| + |(-3)^(-2)^(-1)|

does not exist when using the real numbers only because

|(-1)^(-2)^(-3)| = |(-1)^(-1/2³)| = |(-1)^(-1/8)| = |1/(-1)^(1/8)|,

|(-3)^(-2)^(-1)| = |(-3)^(-1/2)| = |1/(-3)^(1/2)|;

using the complex numbers, we obtain

(-1) ^|^^|+ (-2) ^|^^|+ (-3) = |(-1)^(-2)^(-3)| + |(-1)^(-3)^(-2)| + |(-2)^(-1)^(-3)| + |(-2)^(-3)^(-1)| + |(-3)^(-1)^(-2)| + |(-3)^(-2)^(-1)|

= |(-1)^(-1/8)| + |(-1)^(1/9)| + |(-2)^(-1)| + |(-2)^(-1/3)| + |(-3)^1| + |(-3)^(-1/2)|

= 1 + 1 + 1/2 + 1/2^1/3 + 3 + 1/3^1/2 = 11/2 + 1/2^1/3 + 1/3^1/2 ;

using base-sign-conserving exponentiation denoted by "^ , we obtain

(-1) ^|"^^|+ (-2) ^|"^^|+ (-3) = |(-1)"^(-2)"^(-3)| + |(-1)"^(-3)"^(-2)| + |(-2)"^(-1)"^(-3)| + |(-2)"^(-3)"^(-1)| + |(-3)"^(-1)"^(-2)| + |(-3)"^(-2)"^(-1)|

= |(-1)"^[-2^(-3)]| + |(-1)"^[-3^(-2)]| + |(-2)"^[-1^(-3)]| + |(-2)"^[-3^(-1)]| + |(-3)"^[-1^(-2)]| + |(-3)"^[-2^(-1)]|

= |(-1)"^(-1/8)| + |(-1)"^(-1/9)| + |(-2)"^(-1)| + |(-2)"^(-1/3)| + |(-3)"^(-1)| + |(-3)"^(-1/2)|

= |-1^(-1/8)|+ |-1^(-1/9)| + |-2^(-1)| + |-2^(-1/3)| + |-3^(-1)| + |-3^(-1/2)|

= 1 + 1 + 1/2 + 1/2^1/3 + 1/3 + 1/3^1/2 = 17/6 + 1/2^1/3 + 1/3^1/2 ;

^|^^|+2a = a ^|^^|+ a = a₁ ^|^^|+ a₂ = |a₁^a₂| + |a₂^a₁| = |a^a| + |a^a| = |a^a| + |a^a| = 2|a^a|;

^|^^|+3a = a ^|^^|+ a ^|^^|+ a = a₁ ^|^^|+ a₂ ^|^^|+ a₃

= |a₁^a₂^a₃| + |a₁^a₃^a₂| + |a₂^a₁^a₃| + |a₂^a₃^a₁| + |a₃^a₁^a₂| + |a₃^a₂^a₁|

= |a^a^a| + |a^a^a| + |a^a^a| + |a^a^a| + |a^a^a| + |a^a^a| = 6|a^a^a| = 6|³a|;

^|^^|+2(-2) = (-2) ^|^^|+ (-2) = (-2)₁ ^|^^|+ (-2)₂ = |(-2)₁^(-2)₂| + |(-2)₂^(-2)₁| = |(-2)^(-2)| + |(-2)^(-2)| = 1/2² + 1/2² = 1/2;

^|^^|+3(-2) = (-2) ^|^^|+ (-2) ^|^^|+ (-2) = (-2)₁ ^|^^|+ (-2)₂ ^|^^|+ (-2)₃

= |(-2)₁^(-2)₂^(-2)₃| + |(-2)₁^(-2)₃^(-2)₂| + |(-2)₂^(-2)₁^(-2)₃| + |(-2)₂^(-2)₃^(-2)₁| + |(-2)₃^(-2)₁^(-2)₂| + |(-2)₃^(-2)₂^(-2)₁|

= |(-2)^(-2)^(-2)| + |(-2)^(-2)^(-2)| + |(-2)^(-2)^(-2)| + |(-2)^(-2)^(-2)| + |(-2)^(-2)^(-2)| + |(-2)^(-2)^(-2)|

= 6|(-2)^(-2)^(-2)| = 6|³(-2)| = 6|(-2)^(1/4)| = 6|(-2)^1/4|

does not exist due to (-2)^1/4 when using the real numbers only;

using the complex numbers, we obtain

^|^^|+3(-2) = 6 2^1/4 ;

using base-sign-conserving exponentiation denoted by "^ , we obtain

^|"^^|+3(-2) = (-2) ^|"^^|+ (-2) ^|"^^|+ (-2) = (-2)₁ ^|"^^|+ (-2)₂ ^|"^^|+ (-2)₃

= |(-2)₁"^(-2)₂"^(-2)₃| + |(-2)₁"^(-2)₃"^(-2)₂| + |(-2)₂"^(-2)₁"^(-2)₃| + |(-2)₂"^(-2)₃"^(-2)₁| + |(-2)₃"^(-2)₁"^(-2)₂| + |(-2)₃"^(-2)₂"^(-2)₁|

= |(-2)"^(-2)"^(-2)| + |(-2)"^(-2)"^(-2)| + |(-2)"^(-2)"^(-2)| + |(-2)"^(-2)"^(-2)| + |(-2)"^(-2)"^(-2)| + |(-2)"^(-2)"^(-2)|

= 6|(-2)"^(-2)"^(-2)| = 6|³"(-2)| = 6|(-2)"^[-2^(-2)]| = 6|(-2)"^(-2^-2)| = 6|(-2)"^(-1/4)| = 6|-2^(-1/4)| = 6|-2^-1/4| = 6/2^1/4 .

Commutative Power-Sum-Modulus Exponentiation

Idea

The principal idea of commutative power-sum-modulus exponentiation is as follows:

consider any set of real numbers;

if there are coinciding (equal) numbers among them, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) numbers to artificially differentiate between them;

consider all the permutations of these already distinguishable numbers;

for every permutation of these numbers, compose their power tower using them in the order corresponding to this permutation;

for all these permutations together, compose the modulus (absolute value) of the sum of all these power towers;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative power-sum-modulus exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative power-sum-modulus exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative power-sum-modulus exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative power-sum-modulus exponential is denoted by

... ^^|+| a ^^|+| ... ^^|+| b ^^|+| ... ^^|+| c ^^|+| ... = |... ^⁺ a ^⁺ ... ^⁺ b ^⁺ ... ^⁺ c ^⁺ ...|;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative power-sum-modulus exponential is denoted by

Ε|Σ|_j∈J a_j = |ΕΣ_j∈J a_j| = ^^|+|_j∈J a_j = |^⁺_j∈J a_j| = ^^|+|{a_j | j ∈ J} = |^⁺{a_j | j ∈ J}|

= ... ^^|+| a_j' ^^|+| ... ^^|+| a_j'' ^^|+| ... ^^|+| a_j''' ^^|+| ... = |... ^⁺ a_j' ^⁺ ... ^⁺ a_j'' ^⁺ ... ^⁺ a_j''' ^⁺ ...|;

for any n ∈ N = {1, 2, 3, ...}, a commutative power-sum-modulus tetration is denoted by

^^|+|na = a ^^|+| a ^^|+| ... ^^|+| a = |a ^⁺ a ^⁺ ... ^⁺ a|

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a ^^|+| b = |a ^⁺ b| = |a^b + b^a| = |a^b + b^a|;

a ^^|+| b ^^|+| c = |a ^⁺ b ^⁺ c| = |a^b^c + a^c^b + b^a^c + b^c^a + c^a^b + c^b^a|;

Ε|Σ|_{j∈{1, 2, 3}} a_j = |ΕΣ_{j∈{1, 2, 3}} a_j| = ^^|+|_{j∈{1, 2, 3}} a_j = |^⁺_{j∈{1, 2, 3}} a_j|

= ^^|+|{a_j | j ∈ {1, 2, 3}} = |^⁺{a_j | j ∈ {1, 2, 3}}| = a₁ ^^|+| a₂ ^^|+| a₃ = |a₁ ^⁺ a₂ ^⁺ a₃|

= |a₁^a₂^a₃ + a₁^a₃^a₂ + a₂^a₁^a₃ + a₂^a₃^a₁ + a₃^a₁^a₂ + a₃^a₂^a₁|;

(-2) ^^|+| (-3) = |(-2) ^⁺ (-3)| = |(-2)^(-3) + (-3)^(-2)| = |-1/2^3 + (-1/3^2)| = 1/2³ + 1/3² = 17/72;

(-1) ^^|+| (-2) ^^|+| (-3) = |(-1) ^⁺ (-2) ^⁺ (-3)|

= |(-1)^(-2)^(-3) + (-1)^(-3)^(-2) + (-2)^(-1)^(-3) + (-2)^(-3)^(-1) + (-3)^(-1)^(-2) + (-3)^(-2)^(-1)|

does not exist when using the real numbers only because

(-1)^(-2)^(-3) = (-1)^(-1/2³) = (-1)^(-1/8) = 1/(-1)^(1/8),

(-3)^(-2)^(-1) = (-3)^(-1/2) = 1/(-3)^(1/2);

using base-sign-conserving exponentiation denoted by "^ , we obtain

(-1) "^^|+| (-2) "^^|+| (-3) = |(-1) "^⁺ (-2) "^⁺ (-3)|

= |(-1)"^(-2)"^(-3) + (-1)"^(-3)"^(-2) + (-2)"^(-1)"^(-3) + (-2)"^(-3)"^(-1) + (-3)"^(-1)"^(-2) + (-3)"^(-2)"^(-1)|

= |(-1)"^[-2^(-3)] + (-1)"^[-3^(-2)] + (-2)"^[-1^(-3)] + (-2)"^[-3^(-1)] + (-3)"^[-1^(-2)] + (-3)"^[-2^(-1)]|

= |(-1)"^(-1/8) + (-1)"^(-1/9) + (-2)"^(-1) + (-2)"^(-1/3) + (-3)"^(-1) + (-3)"^(-1/2)|

= |-1^(-1/8) + [-1^(-1/9)] + [-2^(-1)] + [-2^(-1/3)] + [-3^(-1)] + [-3^(-1/2)]|

= |-1 + (-1) + (-1/2) + (-1/2^1/3) + (-1/3) + (-1/3^1/2 )| = 17/6 + 1/2^1/3 + 1/3^1/2 ;

^^|+|2a = |^⁺²a| = a ^^|+| a = |a ^⁺ a| = a₁ ^^|+| a₂ = |a₁ ^⁺ a₂| = |a₁^a₂ + a₂^a₁| = |a^a + a^a| = |a^a + a^a| = 2|a^a|;

^^|+|3a = |^⁺³a| = a ^^|+| a ^^|+| a = |a ^⁺ a ^⁺ a| = a₁ ^^|+| a₂ ^^|+| a₃ = |a₁ ^⁺ a₂ ^⁺ a₃|

= |a₁^a₂^a₃ + a₁^a₃^a₂ + a₂^a₁^a₃ + a₂^a₃^a₁ + a₃^a₁^a₂ + a₃^a₂^a₁|

= |a^a^a + a^a^a + a^a^a + a^a^a + a^a^a + a^a^a| = 6|a^a^a| = 6|³a|;

^^|+|2(-2) = |^⁺²(-2)| = (-2) ^^|+| (-2) = |(-2) ^⁺ (-2)| = (-2)₁ ^^|+| (-2)₂ = |(-2)₁ ^⁺ (-2)₂|

= |(-2)₁^(-2)₂ + (-2)₂^(-2)₁| = |(-2)^(-2) + (-2)^(-2)| = 1/2² + 1/2² = 1/2;

^^|+|3(-2) = |^⁺³(-2)| = (-2) ^^|+| (-2) ^^|+| (-2) = |(-2)₁ ^⁺ (-2)₂ ^⁺ (-2)₃|

= |(-2)₁^(-2)₂^(-2)₃ + (-2)₁^(-2)₃^(-2)₂ + (-2)₂^(-2)₁^(-2)₃ + (-2)₂^(-2)₃^(-2)₁ + (-2)₃^(-2)₁^(-2)₂ + (-2)₃^(-2)₂^(-2)₁|

= |(-2)^(-2)^(-2) + (-2)^(-2)^(-2) + (-2)^(-2)^(-2) + (-2)^(-2)^(-2) + (-2)^(-2)^(-2) + (-2)^(-2)^(-2)|

= 6|(-2)^(-2)^(-2)| = 6|³(-2)| = 6|(-2)^(1/4)| = 6|(-2)^1/4|

does not exist due to (-2)^1/4 when using the real numbers only;

using the complex numbers, we obtain

^^|+|3(-2) = 6 2^1/4 ;

using base-sign-conserving exponentiation denoted by "^ , we obtain

"^^|+|3(-2) = |"^⁺³(-2)| = (-2) "^^|+| (-2) "^^|+| (-2) = |(-2) "^⁺ (-2) "^⁺ (-2)|

= (-2)₁ "^^|+| (-2)₂ "^^|+| (-2)₃ = |(-2)₁ "^⁺ (-2)₂ "^⁺ (-2)₃|

= |(-2)₁"^(-2)₂"^(-2)₃ + (-2)₁"^(-2)₃"^(-2)₂ + (-2)₂"^(-2)₁"^(-2)₃ + (-2)₂"^(-2)₃"^(-2)₁ + (-2)₃"^(-2)₁"^(-2)₂ + (-2)₃"^(-2)₂"^(-2)₁|

= |(-2)"^(-2)"^(-2) + (-2)"^(-2)"^(-2) + (-2)"^(-2)"^(-2) + (-2)"^(-2)"^(-2) + (-2)"^(-2)"^(-2) + (-2)"^(-2)"^(-2)|

= 6|(-2)"^(-2)"^(-2)| = 6|³"(-2)| = 6|(-2)"^[-2^(-2)]| = 6|(-2)"^(-2^-2)| = 6|(-2)"^(-1/4)| = 6|-2^(-1/4)| = 6|-2^-1/4| = 6/2^1/4 .

Commutative Modulus-Power-Sum Exponentiation

Idea

The principal idea of commutative modulus-power-sum exponentiation is as follows:

for any set of real numbers, determine their moduli (absolute values);

if there are coinciding (equal) values among these moduli, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) values to artificially differentiate between them;

consider all the permutations of these already distinguishable values of all those moduli;

for every permutation of these values, compose their power tower using them in the order corresponding to this permutation;

for all these permutations together, compose the sum of all these power towers;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative modulus-power-sum exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative modulus-power-sum exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative modulus-power-sum exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative modulus-power-sum exponential is denoted by

... ^||^⁺ a ^||^⁺ ... ^||^⁺ b ^||^⁺ ... ^||^⁺ c ^||^⁺ ... = ... ^⁺ |a| ^⁺ ... ^⁺ |b| ^⁺ ... ^⁺ |c| ^⁺ ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative modulus-power-sum exponential is denoted by

ΕΣ_j∈J |a_j| = ^||^⁺_j∈J a_j = ^⁺_j∈J |a_j| = ΕΣ{|a_j| | j ∈ J} = ^||^⁺{a_j | j ∈ J} = ^⁺{|a_j| | j ∈ J}

= ... ^||^⁺ a_j' ^||^⁺ ... ^||^⁺ a_j'' ^||^⁺ ... ^||^⁺ a_j''' ^||^⁺ ... = ... ^⁺ |a_j'| ^⁺ ... ^⁺ |a_j''| ^⁺ ... ^⁺ |a_j'''| ^⁺ ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative modulus-power-sum tetration is denoted by

^||^⁺ⁿa = ^⁺ⁿ|a| = a ^||^⁺ a ^||^⁺ ... ^||^⁺ a = |a| ^⁺ |a| ^⁺ ... ^⁺ |a|

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a ^||^⁺ b = |a| ^⁺ |b| = |a|^|b| + |b|^|a| = |a|^|b| + |b|^|a| ;

a ^||^⁺ b ^||^⁺ c = |a| ^⁺ |b| ^⁺ |c| = |a|^|b|^|c| + |a|^|c|^|b| + |b|^|a|^|c| + |b|^|c|^|a| + |c|^|a|^|b| + |c|^|b|^|a|;

ΕΣ_{j∈{1, 2, 3}} |a_j| = ΕΣ{|a_j| | j ∈ {1, 2, 3}}

= ^||^⁺{a_j | j ∈ {1, 2, 3}} = ^||^⁺_{j∈{1, 2, 3}} a_j = ^⁺_{j∈{1, 2, 3}} |a_j| = ^⁺{|a_j| | j ∈ {1, 2, 3}}

= a₁ ^||^⁺ a₂ ^||^⁺ a₃ = |a₁| ^⁺ |a₂| ^⁺ |a₃|

= |a₁|^|a₂|^|a₃| + |a₁|^|a₃|^|a₂| + |a₂|^|a₁|^|a₃| + |a₂|^|a₃|^|a₁| + |a₃|^|a₁|^|a₂| + |a₃|^|a₂|^|a₁| ;

(-2) ^||^⁺ (-3) = |-2| ^⁺ |-3| = 2 ^||^⁺ 3 = 2 ^⁺ 3 = |-2|^|-3| + |-3|^|-2| = 2^3 + 3^2 = 2³ + 3² = 17;

(-1) ^||^⁺ (-2) ^||^⁺ (-3) = |-1| ^⁺ |-2| ^⁺ |-3| = 1 ^||^⁺ 2 ^||^⁺ 3 = 1 ^⁺ 2 ^⁺ 3

= 1^2^3 + 1^3^2 + 2^1^3 + 2^3^1 + 3^1^2 + 3^2^1

= 1^8 + 1^9 + 2^1 + 2^3 + 3^1 + 3^2 = 1⁸ + 1⁹ + 2¹ + 2³ + 3¹ + 3² = 24;

^||^⁺²a = ^⁺²|a| = a ^||^⁺ a = |a| ^⁺ |a| = a₁ ^||^⁺ a₂ = |a₁| ^⁺ |a₂|

= |a₁|^|a₂| + |a₂|^|a₁|

= |a|^|a| + |a|^|a| = |a|^|a| + |a|^|a| = 2|a|^|a| ;

^||^⁺³a = ^⁺³|a| = a ^||^⁺ a ^||^⁺ a = |a| ^⁺ |a| ^⁺ |a| = a₁ ^||^⁺ a₂ ^||^⁺ a₃ = |a₁| ^⁺ |a₂| ^⁺ |a₃|

= |a₁|^|a₂|^|a₃| + |a₁|^|a₃|^|a₂| + |a₂|^|a₁|^|a₃| + |a₂|^|a₃|^|a₁| + |a₃|^|a₁|^|a₂| + |a₃|^|a₂|^|a₁|

= |a|^|a|^|a| + |a|^|a|^|a| + |a|^|a|^|a| + |a|^|a|^|a| + |a|^|a|^|a| + |a|^|a|^|a| = 6|a|^|a|^|a| = 6 ³|a|;

^||^⁺²(-2) = ^⁺²|-2| = (-2) ^||^⁺ (-2) = |-2|^⁺ |-2| = 2 ^⁺ 2 = 2₁ ^⁺ 2₂ = 2₁^2₂ + 2₂^2₁ = 2^2 + 2^2 = 2² + 2² = 8;

^||^⁺³(-2) = ^⁺³|-2| = (-2) ^||^⁺ (-2) ^||^⁺ (-2) = |-2| ^⁺ |-2| ^⁺ |-2| = 2 ^⁺ 2 ^⁺ 2 = 2₁ ^⁺ 2₂ ^⁺ 2₃

= 2₁^2₂^2₃ + 2₁^2₃^2₂ + 2₂^2₁^2₃ + 2₂^2₃^2₁ + 2₃^2₁^2₂ + 2₃^2₂^2₁

= 2^2^2 + 2^2^2 + 2^2^2 + 2^2^2 + 2^2^2 + 2^2^2

= 2^4 + 2^4 + 2^4 + 2^4 + 2^4 + 2^4 = 2⁴ + 2⁴ + 2⁴ + 2⁴ + 2⁴ + 2⁴ = 96.

Commutative Sign-Power-Sum Exponentiation Mode

Commutative Sign-Power-Modulus-Sum Exponentiation

Idea

The principal idea of commutative sign-power-modulus-sum exponentiation is as follows:

consider any set of real numbers;

if there are coinciding (equal) numbers among them, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) numbers to artificially differentiate between them;

consider all the permutations of these already distinguishable numbers;

for every permutation of these numbers, compose their power tower using them in the order corresponding to this permutation;

for every permutation of these numbers, multiply the modulus of this power tower with its base sign;

for all these permutations together, compose the sum of the products of the moduli (absolute values) of all these power towers with their bases signs;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative sign-power-modulus-sum exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative sign-power-modulus-sum exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative sign-power-modulus-sum exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative sign-power-modulus-sum exponential is denoted by

... °^|^^|+ a °^|^^|+ ... °^|^^|+ b °^|^^|+ ... °^|^^|+ c °^|^^|+ ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative sign-power-modulus-sum exponential is denoted by

°|Ε|Σ_j∈J a_j = °^|^^|+_j∈J a_j = °^|^^|+{a_j | j ∈ J}

= ... °^|^^|+ a_j' °^|^^|+ ... °^|^^|+ a_j'' °^|^^|+ ... °^|^^|+ a_j''' °^|^^|+ ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative sign-power-modulus-sum tetration is denoted by

°^|^^|+na = a °^|^^|+ a °^|^^|+ ... °^|^^|+ a

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a °^|^^|+ b = a°|a^b| + b°|b^a| = a°|a^b| + b°|b^a| = (sign a)|a^b| + (sign b)|b^a| = (sign a)|a^b| + (sign b)|b^a|;

a °^|^^|+ b °^|^^|+ c = a°|a^b^c| + a°|a^c^b| + b°|b^a^c| + b°|b^c^a| + c°|c^a^b| + c°|c^b^a|

= (sign a)|a^b^c| + (sign a)|a^c^b| + (sign b)|b^a^c| + (sign b)|b^c^a| + (sign c)|c^a^b| + (sign c)|c^b^a|;

°|Ε|Σ_{j∈{1, 2, 3}} a_j = °^|^^|+_{j∈{1, 2, 3}} a_j = °^|^^|+{a_j | j ∈ {1, 2, 3}} = a₁ °^|^^|+ a₂ °^|^^|+ a₃

= a₁°|a₁^a₂^a₃| + a₁°|a₁^a₃^a₂| + a₂°|a₂^a₁^a₃| + a₂°|a₂^a₃^a₁| + a₃°|a₃^a₁^a₂| + a₃°|a₃^a₂^a₁|

= (sign a₁)|a₁^a₂^a₃| + (sign a₁)|a₁^a₃^a₂| + (sign a₂)|a₂^a₁^a₃| + (sign a₂)|a₂^a₃^a₁| + (sign a₃)|a₃^a₁^a₂| + (sign a₃)|a₃^a₂^a₁|;

(-2) °^|^^|+ (-3) = (-2)°|(-2)^(-3)| + (-3)°|(-3)^(-2)| = sign(-2)|(-2)^(-3)| + sign(-3)|(-3)^(-2)|

= (-1)|-1/2^3| + (-1)|-1/3^2| = (-1)1/2³ + (-1)1/3² = -17/72;

(-1) °^|^^|+ (-2) °^|^^|+ (-3)

= (-1)°|(-1)^(-2)^(-3)| + (-1)°|(-1)^(-3)^(-2)| + (-2)°|(-2)^(-1)^(-3)| + (-2)°|(-2)^(-3)^(-1)| + (-3)°|(-3)^(-1)^(-2)| + (-3)°|(-3)^(-2)^(-1)|

= (-1)|(-1)^(-2)^(-3)| + (-1)|(-1)^(-3)^(-2)| + (-1)|(-2)^(-1)^(-3)| + (-1)|(-2)^(-3)^(-1)| + (-1)|(-3)^(-1)^(-2)| + (-1)|(-3)^(-2)^(-1)|

does not exist when using the real numbers only because

|(-1)^(-2)^(-3)| = |(-1)^(-1/2³)| = |(-1)^(-1/8)| = |1/(-1)^(1/8)|,

|(-3)^(-2)^(-1)| = |(-3)^(-1/2)| = |1/(-3)^(1/2)|;

using the complex numbers, we obtain

(-1) °^|^^|+ (-2) °^|^^|+ (-3)

= (-1)°|(-1)^(-2)^(-3)| + (-1)°|(-1)^(-3)^(-2)| + (-2)°|(-2)^(-1)^(-3)| + (-2)°|(-2)^(-3)^(-1)| + (-3)°|(-3)^(-1)^(-2)| + (-3)°|(-3)^(-2)^(-1)|

= sign(-1)|(-1)^(-1/8)| + sign(-1)|(-1)^(1/9)| + sign(-2)|(-2)^(-1)| + sign(-2)|(-2)^(-1/3)| + sign(-3)|(-3)^1| + sign(-3)|(-3)^(-1/2)|

= (-1)1 + (-1)1 + (-1)1/2 + (-1)1/2^1/3 + (-1)3 + (-1)1/3^1/2 = - 11/2 - 1/2^1/3 - 1/3^1/2 ;

using base-sign-conserving exponentiation denoted by "^ , we obtain

(-1) °^|"^^|+ (-2) °^|"^^|+ (-3)

= (-1)°|(-1)"^(-2)"^(-3)| + (-1)°|(-1)"^(-3)"^(-2)| + (-2)°|(-2)"^(-1)"^(-3)| + (-2)°|(-2)"^(-3)"^(-1)| + (-3)°|(-3)"^(-1)"^(-2)| + (-3)°|(-3)"^(-2)"^(-1)|

= (-1)|(-1)"^[-2^(-3)]| + (-1)|(-1)"^[-3^(-2)]| + (-1)|(-2)"^[-1^(-3)]| + (-1)|(-2)"^[-3^(-1)]| + (-1)|(-3)"^[-1^(-2)]| + (-1)|(-3)"^[-2^(-1)]|

= sign(-1)|(-1)"^(-1/8)| + sign(-1)|(-1)"^(-1/9)| + sign(-2)|(-2)"^(-1)| + sign(-2)|(-2)"^(-1/3)| + sign(-3)|(-3)"^(-1)| + sign(-3)|(-3)"^(-1/2)|

= (-1)|-1^(-1/8)| + (-1)|-1^(-1/9)| + (-1)|-2^(-1)| + (-1)|-2^(-1/3)| + (-1)|-3^(-1)| + (-1)|-3^(-1/2)|

= (-1)1 + (-1)1 + (-1)1/2 + (-1)1/2^1/3 + (-1)1/3 + (-1)1/3^1/2 = - 17/6 - 1/2^1/3 - 1/3^1/2 ;

°^|^^|+2a = a °^|^^|+ a

= a₁ °^|^^|+ a₂ = a₁°|a₁^a₂| + a₂°|a₂^a₁| = (sign a₁)|a₁^a₂| + (sign a₂)|a₂^a₁|

= a°|a^a| + a°|a^a| = a°|a^a| + a°|a^a| = 2a°|a^a|;

°^|^^|+3a = a °^|^^|+ a °^|^^|+ a = a₁ °^|^^|+ a₂ °^|^^|+ a₃

= a₁°|a₁^a₂^a₃| + a₁°|a₁^a₃^a₂| + a₂°|a₂^a₁^a₃| + a₂°|a₂^a₃^a₁| + a₃°|a₃^a₁^a₂| + a₃°|a₃^a₂^a₁|

= (sign a₁)|a₁^a₂^a₃| + (sign a₁)|a₁^a₃^a₂| + (sign a₂)|a₂^a₁^a₃| + (sign a₂)|a₂^a₃^a₁| + (sign a₃)|a₃^a₁^a₂| + (sign a₃)|a₃^a₂^a₁|

= a°|a^a^a| + a°|a^a^a| + a°|a^a^a| + a°|a^a^a| + a°|a^a^a| + a°|a^a^a|

= (sign a)|a^a^a| + (sign a)|a^a^a| + (sign a)|a^a^a| + (sign a)|a^a^a| + (sign a)|a^a^a| + (sign a)|a^a^a|

= 6a°|a^a^a| = 6(sign a)|a^a^a| = 6a°|³a|;

°^|^^|+2(-2) = (-2) °^|^^|+ (-2) = (-2)₁ °^|^^|+ (-2)₂

= (-2)₁°|(-2)₁^(-2)₂| + (-2)₂°|(-2)₂^(-2)₁| = sign(-2)₁|(-2)₁^(-2)₂| + sign(-2)₂|(-2)₂^(-2)₁|

= (-2)°|(-2)^(-2)| + (-2)°|(-2)^(-2)| = sign(-2)|(-2)^(-2)| + sign(-2)|(-2)^(-2)| = (-1)1/2² + (-1)1/2² = -1/2;

°^|^^|+3(-2) = (-2) °^|^^|+ (-2) °^|^^|+ (-2) = (-2)₁ °^|^^|+ (-2)₂ °^|^^|+ (-2)₃

= (-2)₁°|(-2)₁^(-2)₂^(-2)₃| + (-2)₁°|(-2)₁^(-2)₃^(-2)₂| + (-2)₂°|(-2)₂^(-2)₁^(-2)₃|

+ (-2)₂°|(-2)₂^(-2)₃^(-2)₁| + (-2)₃°|(-2)₃^(-2)₁^(-2)₂| + (-2)₃°|(-2)₃^(-2)₂^(-2)₁|

= (-2)°|(-2)^(-2)^(-2)| + (-2)°|(-2)^(-2)^(-2)| + (-2)°|(-2)^(-2)^(-2)| + (-2)°|(-2)^(-2)^(-2)| + (-2)°|(-2)^(-2)^(-2)| + (-2)°|(-2)^(-2)^(-2)|

= 6(-2)°|(-2)^(-2)^(-2)| = 6(-1)|(-2)^(-2)^(-2)| = 6(-1)|³(-2)| = 6(-1)|(-2)^(1/4)| = 6(-1)|(-2)^1/4|

does not exist due to (-2)^1/4 when using the real numbers only;

using the complex numbers, we obtain

°^|^^|+3(-2) = - 6 2^1/4 ;

using base-sign-conserving exponentiation denoted by "^ , we obtain

°^|"^^|+3(-2) = (-2) °^|"^^|+ (-2) °^|"^^|+ (-2) = (-2)₁ °^|"^^|+ (-2)₂ °^|"^^|+ (-2)₃

= (-2)₁°|(-2)₁"^(-2)₂"^(-2)₃| + (-2)₁°|(-2)₁"^(-2)₃"^(-2)₂| + (-2)₂°|(-2)₂"^(-2)₁"^(-2)₃|

+ (-2)₂°|(-2)₂"^(-2)₃"^(-2)₁| + (-2)₃°|(-2)₃"^(-2)₁"^(-2)₂| + (-2)₃°|(-2)₃"^(-2)₂"^(-2)₁|

= (-2)°|(-2)"^(-2)"^(-2)| + (-2)°|(-2)"^(-2)"^(-2)| + (-2)°|(-2)"^(-2)"^(-2)| + (-2)°|(-2)"^(-2)"^(-2)| + (-2)°|(-2)"^(-2)"^(-2)| + (-2)°|(-2)"^(-2)"^(-2)|

= 6(-2)°|(-2)"^(-2)"^(-2)| = 6(-2)°|³"(-2)| = 6(-2)°|(-2)"^[-2^(-2)]|

= 6(-2)°|(-2)"^(-2^-2)| = 6(-2)°|(-2)"^(-1/4)| = 6(-2)°|-2^(-1/4)| = 6(-1)|-2^-1/4| = - 6/2^1/4 .

Commutative Sign-Power-Sum-Modulus Exponentiation

Idea

The principal idea of commutative sign-power-sum-modulus exponentiation is as follows:

consider any set of real numbers;

if there are coinciding (equal) numbers among them, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) numbers to artificially differentiate between them;

consider all the permutations of these already distinguishable numbers;

for every permutation of these numbers, compose their power tower using them in the order corresponding to this permutation;

for every permutation of these numbers, multiply the modulus of this power tower with its base sign;

for all these permutations together, compose the modulus of the sum of the products of the moduli (absolute values) of all these power towers with their bases signs;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative sign-power-sum-modulus exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative sign-power-sum-modulus exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative sign-power-sum-modulus exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative sign-power-sum-modulus exponential is denoted by

... °^|^^+| a °^|^^+| ... °^|^^+| b °^|^^+| ... °^|^^+| c °^|^^+| ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative sign-power-sum-modulus exponential is denoted by

°|ΕΣ|_j∈J a_j = °^|^^|+_j∈J a_j = °^|^^|+{a_j | j ∈ J}

= ... °^|^^|+ a_j' °^|^^|+ ... °^|^^|+ a_j'' °^|^^|+ ... °^|^^|+ a_j''' °^|^^|+ ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative sign-power-sum-modulus tetration is denoted by

°^|^^|+na = a °^|^^|+ a °^|^^|+ ... °^|^^|+ a

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a °^|^^+| b = |a°|a^b| + b°|b^a|| = |(sign a)|a^b| + (sign b)|b^a||;

a °^|^^+| b = |a°|a^b| + b°|b^a|| = |a°|a^b| + b°|b^a|| = |(sign a)|a^b| + (sign b)|b^a|| = |(sign a)|a^b| + (sign b)|b^a||;

a °^|^^+| b °^|^^+| c = |a°|a^b^c| + a°|a^c^b| + b°|b^a^c| + b°|b^c^a| + c°|c^a^b| + c°|c^b^a||

= |(sign a)|a^b^c| + (sign a)|a^c^b| + (sign b)|b^a^c| + (sign b)|b^c^a| + (sign c)|c^a^b| + (sign c)|c^b^a||;

°|ΕΣ|_{j∈{1, 2, 3}} a_j = °^|^^+|_{j∈{1, 2, 3}} a_j = °^|^^+|{a_j | j ∈ {1, 2, 3}} = a₁ °^|^^+| a₂ °^|^^+| a₃

= |a₁°|a₁^a₂^a₃| + a₁°|a₁^a₃^a₂| + a₂°|a₂^a₁^a₃| + a₂°|a₂^a₃^a₁| + a₃°|a₃^a₁^a₂| + a₃°|a₃^a₂^a₁||

= |(sign a₁)|a₁^a₂^a₃| + (sign a₁)|a₁^a₃^a₂| + (sign a₂)|a₂^a₁^a₃| + (sign a₂)|a₂^a₃^a₁| + (sign a₃)|a₃^a₁^a₂| + (sign a₃)|a₃^a₂^a₁||;

(-2) °^|^^+| (-3) = |(-2)°|(-2)^(-3)| + (-3)°|(-3)^(-2)|| = |sign(-2)|(-2)^(-3)| + sign(-3)|(-3)^(-2)||

= |(-1)|-1/2^3| + (-1)|-1/3^2|| = |(-1)1/2³ + (-1)1/3²| = |-17/72| = 17/72;

(-1) °^|^^+| (-2) °^|^^+| (-3)

= |(-1)°|(-1)^(-2)^(-3)| + (-1)°|(-1)^(-3)^(-2)| + (-2)°|(-2)^(-1)^(-3)| + (-2)°|(-2)^(-3)^(-1)| + (-3)°|(-3)^(-1)^(-2)| + (-3)°|(-3)^(-2)^(-1)||

= |(-1)|(-1)^(-2)^(-3)| + (-1)|(-1)^(-3)^(-2)| + (-1)|(-2)^(-1)^(-3)| + (-1)|(-2)^(-3)^(-1)| + (-1)|(-3)^(-1)^(-2)| + (-1)|(-3)^(-2)^(-1)||

does not exist when using the real numbers only because

|(-1)^(-2)^(-3)| = |(-1)^(-1/2³)| = |(-1)^(-1/8)| = |1/(-1)^(1/8)|,

|(-3)^(-2)^(-1)| = |(-3)^(-1/2)| = |1/(-3)^(1/2)|;

using the complex numbers, we obtain

(-1) °^|^^+| (-2) °^|^^+| (-3)

= |(-1)°|(-1)^(-2)^(-3)| + (-1)°|(-1)^(-3)^(-2)| + (-2)°|(-2)^(-1)^(-3)| + (-2)°|(-2)^(-3)^(-1)| + (-3)°|(-3)^(-1)^(-2)| + (-3)°|(-3)^(-2)^(-1)||

= |sign(-1)|(-1)^(-1/8)| + sign(-1)|(-1)^(1/9)| + sign(-2)|(-2)^(-1)| + sign(-2)|(-2)^(-1/3)| + sign(-3)|(-3)^1| + sign(-3)|(-3)^(-1/2)||

= |(-1)1 + (-1)1 + (-1)1/2 + (-1)1/2^1/3 + (-1)3 + (-1)1/3^1/2| = |- 11/2 - 1/2^1/3 - 1/3^1/2| = 11/2 + 1/2^1/3 + 1/3^1/2 ;

using base-sign-conserving exponentiation denoted by "^ , we obtain

(-1) °^|"^^+| (-2) °^|"^^+| (-3)

= |(-1)°|(-1)"^(-2)"^(-3)| + (-1)°|(-1)"^(-3)"^(-2)| + (-2)°|(-2)"^(-1)"^(-3)| + (-2)°|(-2)"^(-3)"^(-1)| + (-3)°|(-3)"^(-1)"^(-2)| + (-3)°|(-3)"^(-2)"^(-1)||

= |(-1)|(-1)"^[-2^(-3)]| + (-1)|(-1)"^[-3^(-2)]| + (-1)|(-2)"^[-1^(-3)]| + (-1)|(-2)"^[-3^(-1)]| + (-1)|(-3)"^[-1^(-2)]| + (-1)|(-3)"^[-2^(-1)]||

= |sign(-1)|(-1)"^(-1/8)| + sign(-1)|(-1)"^(-1/9)| + sign(-2)|(-2)"^(-1)| + sign(-2)|(-2)"^(-1/3)| + sign(-3)|(-3)"^(-1)| + sign(-3)|(-3)"^(-1/2)||

= |(-1)|-1^(-1/8)| + (-1)|-1^(-1/9)| + (-1)|-2^(-1)| + (-1)|-2^(-1/3)| + (-1)|-3^(-1)| + (-1)|-3^(-1/2)||

= |(-1)1 + (-1)1 + (-1)1/2 + (-1)1/2^1/3 + (-1)1/3 + (-1)1/3^1/2| = |- 17/6 - 1/2^1/3 - 1/3^1/2| = 17/6 + 1/2^1/3 + 1/3^1/2 ;

°^|^^+|2a = a °^|^^+| a

= a₁ °^|^^+| a₂ = |a₁°|a₁^a₂| + a₂°|a₂^a₁|| = |(sign a₁)|a₁^a₂| + (sign a₂)|a₂^a₁||

= |a°|a^a| + a°|a^a|| = |a°|a^a| + a°|a^a|| = |2a°|a^a|| = 2|a^a|;

°^|^^+|3a = a °^|^^+| a °^|^^+| a = a₁ °^|^^+| a₂ °^|^^+| a₃

= |a₁°|a₁^a₂^a₃| + a₁°|a₁^a₃^a₂| + a₂°|a₂^a₁^a₃| + a₂°|a₂^a₃^a₁| + a₃°|a₃^a₁^a₂| + a₃°|a₃^a₂^a₁||

= |(sign a₁)|a₁^a₂^a₃| + (sign a₁)|a₁^a₃^a₂| + (sign a₂)|a₂^a₁^a₃| + (sign a₂)|a₂^a₃^a₁| + (sign a₃)|a₃^a₁^a₂| + (sign a₃)|a₃^a₂^a₁||

= |a°|a^a^a| + a°|a^a^a| + a°|a^a^a| + a°|a^a^a| + a°|a^a^a| + a°|a^a^a||

= |(sign a)|a^a^a| + (sign a)|a^a^a| + (sign a)|a^a^a| + (sign a)|a^a^a| + (sign a)|a^a^a| + (sign a)|a^a^a||

= |6a°|a^a^a|| = |6(sign a)|a^a^a|| = |6a°|³a|| = 6|³a|;

°^|^^+|2(-2) = (-2) °^|^^+| (-2) = (-2)₁ °^|^^+| (-2)₂

= |(-2)₁°|(-2)₁^(-2)₂| + (-2)₂°|(-2)₂^(-2)₁|| = |sign(-2)₁|(-2)₁^(-2)₂| + sign(-2)₂|(-2)₂^(-2)₁||

= |(-2)°|(-2)^(-2)| + (-2)°|(-2)^(-2)|| = |sign(-2)|(-2)^(-2)| + sign(-2)|(-2)^(-2)|| = |(-1)1/2² + (-1)1/2²| = |-1/2| = 1/2;

°^|^^+|3(-2) = (-2) °^|^^+| (-2) °^|^^+| (-2) = (-2)₁ °^|^^+| (-2)₂ °^|^^+| (-2)₃

= |(-2)₁°|(-2)₁^(-2)₂^(-2)₃| + (-2)₁°|(-2)₁^(-2)₃^(-2)₂| + (-2)₂°|(-2)₂^(-2)₁^(-2)₃|

+ (-2)₂°|(-2)₂^(-2)₃^(-2)₁| + (-2)₃°|(-2)₃^(-2)₁^(-2)₂| + (-2)₃°|(-2)₃^(-2)₂^(-2)₁||

= |(-2)°|(-2)^(-2)^(-2)| + (-2)°|(-2)^(-2)^(-2)| + (-2)°|(-2)^(-2)^(-2)| + (-2)°|(-2)^(-2)^(-2)| + (-2)°|(-2)^(-2)^(-2)| + (-2)°|(-2)^(-2)^(-2)||

= |6(-2)°|(-2)^(-2)^(-2)|| = |6(-1)|(-2)^(-2)^(-2)|| = |6(-1)|³(-2)|| = |6(-1)|(-2)^(1/4)|| = |6(-1)|(-2)^1/4|| = 6|(-2)^1/4|

does not exist due to (-2)^1/4 when using the real numbers only;

using the complex numbers, we obtain

°^|^^+|3(-2) = 6 2^1/4 ;

using base-sign-conserving exponentiation denoted by "^ , we obtain

°^|"^^+|3(-2) = (-2) °^|^^+| (-2) °^|^^+| (-2) = (-2)₁ °^|^^+| (-2)₂ °^|^^+| (-2)₃

= |(-2)₁°|(-2)₁"^(-2)₂"^(-2)₃| + (-2)₁°|(-2)₁"^(-2)₃"^(-2)₂| + (-2)₂°|(-2)₂"^(-2)₁"^(-2)₃|

+ (-2)₂°|(-2)₂"^(-2)₃"^(-2)₁| + (-2)₃°|(-2)₃"^(-2)₁"^(-2)₂| + (-2)₃°|(-2)₃"^(-2)₂"^(-2)₁||

= |(-2)°|(-2)"^(-2)"^(-2)| + (-2)°|(-2)"^(-2)"^(-2)| + (-2)°|(-2)"^(-2)"^(-2)| + (-2)°|(-2)"^(-2)"^(-2)| + (-2)°|(-2)"^(-2)"^(-2)| + (-2)°|(-2)"^(-2)"^(-2)||

= |6(-2)°|(-2)"^(-2)"^(-2)|| = |6(-2)°|³"(-2)|| = |6(-2)°|(-2)"^[-2^(-2)]||

= |6(-2)°|(-2)"^(-2^-2)|| = |6(-2)°|(-2)"^(-1/4)|| = |6(-2)°|-2^(-1/4)|| = |6(-1)|-2^-1/4|| = |- 6/2^1/4| = 6/2^1/4 .

Commutative Sign-Modulus-Power-Sum Exponentiation

Idea

The principal idea of commutative sign-modulus-power-sum exponentiation is as follows:

for any set of real numbers, determine their moduli (absolute values);

if there are coinciding (equal) values among these moduli, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) values to artificially differentiate between them;

consider all the permutations of these already distinguishable values of all those moduli;

for every permutation of these moduli, compose their power tower using them in the order corresponding to this permutation;

for every permutation of these moduli, assign (attach) this power tower the initial sign of the number whose modulus is the base of this power tower;

for all these permutations together, compose the sum of all these power towers with these signs;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative sign-modulus-power-sum exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative sign-modulus-power-sum exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative sign-modulus-power-sum exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative sign-modulus-power-sum exponential is denoted by

... °^||^⁺ a °^||^⁺ ... °^||^⁺ b °^||^⁺ ... °^||^⁺ c °^||^⁺ ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative sign-modulus-power-sum exponential is denoted by

°||ΕΣ_j∈J a_j = °^||^⁺_j∈J a_j = °^||^⁺{a_j | j ∈ J}

= ... °^||^⁺ a_j' °^||^⁺ ... °^||^⁺ a_j'' °^||^⁺ ... °^||^⁺ a_j''' °^||^⁺ ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative sign-modulus-power-sum tetration is denoted by

°^||^⁺ⁿa = a °^||^⁺ a °^||^⁺ ... °^||^⁺ a

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a °^||^⁺ b = a°|a|^|b| + b°|b|^|a| = (sign a)|a|^|b| + (sign b)|b|^|a|

= a°|a|^|b| + b°|b|^|a| = (sign a)|a|^|b| + (sign b)|b|^|a|;

a °^||^⁺ b °^||^⁺ c = a°|a|^|b|^|c| + a°|a|^|c|^|b| + b°|b|^|a|^|c| + b°|b|^|c|^|a| + c°|c|^|a|^|b| + c°|c|^|b|^|a|

= (sign a)|a|^|b|^|c| + (sign a)|a|^|c|^|b| + (sign b)|b|^|a|^|c| + (sign b)|b|^|c|^|a| + (sign c)|c|^|a|^b| + (sign c)|c|^|b|^|a|;

°||ΕΣ_{j∈{1, 2, 3}} a_j = °^||^⁺_{j∈{1, 2, 3}} a_j = °^||^⁺{a_j | j ∈ {1, 2, 3}} = a₁ °^||^⁺ a₂ °^||^⁺ a₃

= a₁°|a₁|^|a₂|^|a₃| + a₁°|a₁|^|a₃|^|a₂| + a₂°|a₂|^|a₁|^|a₃| + a₂°|a₂|^|a₃|^|a₁| + a₃°|a₃|^|a₁|^|a₂| + a₃°|a₃|^|a₂|^|a₁|

= (sign a₁)|a₁|^|a₂|^|a₃| + (sign a₁)|a₁|^|a₃|^|a₂| + (sign a₂)|a₂|^|a₁|^|a₃|

+ (sign a₂)|a₂|^|a₃|^|a₁| + (sign a₃)|a₃|^|a₁|^|a₂| + (sign a₃)|a₃|^|a₂|^|a₁|;

(-2) °^||^⁺ (-3) = (-2)°|-2|^|-3| + (-3)°|-3|^|-2| = sign(-2)|-2|^|-3| + sign(-3)|-3|^|-2|

= (-1)2^3 + (-1)3^2 = (-1)2³ + (-1)3² = -17;

(-1) °^||^⁺ (-2) °^||^⁺ (-3)

= (-1)°|-1|^|-2|^|-3| + (-1)°|-1|^|-3|^|-2| + (-2)°|-2|^|-1|^|-3| + (-2)°|-2|^|-3|^|-1| + (-3)°|-3|^|-1|^|-2| + (-3)°|-3|^|-2|^|-1|

= (-1)|-1|^|-2|^|-3| + (-1)|-1|^|-3|^|-2| + (-1)|-2|^|-1|^|-3| + (-1)|-2|^|-3|^|-1| + (-1)|-3|^|-1|^|-2| + (-1)|-3|^|-2|^|-1|

= (-1)(1^2^3 + 1^3^2 + 2^1^3 + 2^3^1 + 3^1^2 + 3^2^1)

= (-1)(1^8 + 1^9 + 2^1 + 2^3 + 3^1 + 3^2) = (-1)(1⁸ + 1⁹ + 2¹ + 2³ + 3¹ + 3²) = -24;

°^||^⁺²a = a °^||^⁺ a

= a₁ °^||^⁺ a₂ = a₁°|a₁|^|a₂| + a₂°|a₂|^|a₁| = (sign a₁)|a₁|^|a₂| + (sign a₂)|a₂|^|a₁|

= a°|a|^|a| + a°|a|^|a| = a°|a|^|a| + a°|a|^|a| = 2a°|a|^|a| ;

°^||^⁺³a = a °^||^⁺ a °^||^⁺ a = a₁ °^||^⁺ a₂ °^||^⁺ a₃

= a₁°|a₁|^|a₂|^|a₃| + a₁°|a₁|^|a₃|^|a₂| + a₂°|a₂|^|a₁|^|a₃| + a₂°|a₂|^|a₃|^|a₁| + a₃°|a₃|^|a₁|^|a₂| + a₃°|a₃|^|a₂|^|a₁|

= (sign a₁)|a₁|^|a₂|^|a₃| + (sign a₁)|a₁|^|a₃|^|a₂| + (sign a₂)|a₂|^|a₁|^|a₃|

+ (sign a₂)|a₂|^|a₃|^|a₁| + (sign a₃)|a₃|^|a₁|^|a₂| + (sign a₃)|a₃|^|a₂|^|a₁|

= a°|a|^|a|^|a| + a°|a|^|a|^|a| + a°|a|^|a|^|a| + a°|a|^|a|^|a| + a°|a|^|a|^|a| + a°|a|^|a|^|a|

= (sign a)|a|^|a|^|a| + (sign a)|a|^|a|^|a| + (sign a)|a|^|a|^|a| + (sign a)|a|^|a|^|a| + (sign a)|a|^|a|^|a| + (sign a)|a|^|a|^|a|

= 6a°|a|^|a|^|a| = 6(sign a)|a|^|a|^|a| = 6a°³|a|;

°^||^⁺²(-2) = (-2) °^||^⁺ (-2) = (-2)₁ °^||^⁺ (-2)₂

= (-2)₁°|(-2)₁|^|(-2)₂| + (-2)₂°|(-2)₂|^|(-2)₁| = sign(-2)₁|(-2)₁|^|(-2)₂| + sign(-2)₂|(-2)₂|^|(-2)₁|

= (-2)°|-2|^|-2| + (-2)°|-2|^|-2| = sign(-2)|-2|^|-2| + sign(-2)|-2|^|-2| = (-1)2² + (-1)2² = -8;

°^||^⁺³(-2) = (-2) °^||^⁺ (-2) °^||^⁺ (-2) = (-2)₁ °^||^⁺ (-2)₂ °^||^⁺ (-2)₃

= (-2)₁°|(-2)₁|^|(-2)₂|^|(-2)₃| + (-2)₁°|(-2)₁|^|(-2)₃|^|(-2)₂| + (-2)₂°|(-2)₂|^|(-2)₁|^|(-2)₃|

+ (-2)₂°|(-2)₂|^|(-2)₃|^|(-2)₁| + (-2)₃°|(-2)₃|^|(-2)₁|^|(-2)₂| + (-2)₃°|(-2)₃|^|(-2)₂|^|(-2)₁|

= (-2)°|-2|^|-2|^|-2| + (-2)°|-2|^|-2|^|-2| + (-2)°|-2|^|-2|^|-2| + (-2)°|-2|^|-2|^|-2| + (-2)°|-2|^|-2|^|-2| + (-2)°|-2|^|-2|^|-2|

= 6(-2)°|-2|^|-2|^|-2| = 6(-1)|-2|^|-2|^|-2| = 6(-1)³|-2| = 6(-1)³2 = 6(-1)16 = -96.

Commutative Modulus-Sign-Power-Sum Exponentiation

Idea

The principal idea of commutative modulus-sign-power-sum exponentiation is as follows:

for any set of real numbers, determine their moduli (absolute values);

if there are coinciding (equal) values among these moduli, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) values to artificially differentiate between them;

consider all the permutations of these already distinguishable values of all those moduli;

for every permutation of these moduli, compose their power tower using them in the order corresponding to this permutation;

for every permutation of these moduli, assign (attach) this power tower the initial sign of the number whose modulus is the base of this power tower;

for all these permutations together, compose the modulus of the sum of all these power towers with these signs;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative modulus-sign-power-sum exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative modulus-sign-power-sum exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative modulus-sign-power-sum exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative modulus-sign-power-sum exponential is denoted by

... ^|°^^+| a ^|°^^+| ... ^|°^^+| b ^|°^^+| ... ^|°^^+| c ^|°^^+| ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative modulus-sign-power-sum exponential is denoted by

|°ΕΣ|_j∈J a_j = ^|°^^+|_j∈J a_j = ^|°^^+|{a_j | j ∈ J}

= ... ^|°^^+| a_j' ^|°^^+| ... ^|°^^+| a_j'' ^|°^^+| ... ^|°^^+| a_j''' ^|°^^+| ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative modulus-sign-power-sum tetration is denoted by

^|°^^+|na = a ^|°^^+| a ^|°^^+| ... ^|°^^+| a

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a ^|°^^+| b = |a°|a|^|b| + b°|b|^|a|| = |(sign a) |a|^|b| + (sign b) |b|^|a||;

a ^|°^^+| b = |a°|a|^|b| + b°|b|^|a|| = |(sign a)|a|^|b| + (sign b)|b|^|a||

= |a°|a|^|b| + b°|b|^|a|| = |(sign a)|a|^|b| + (sign b)|b|^|a||;

a ^|°^^+| b ^|°^^+| c = |a°|a|^|b|^|c| + a°|a|^|c|^|b| + b°|b|^|a|^|c| + b°|b|^|c|^|a| + c°|c|^|a|^|b| + c°|c|^|b|^|a||

= |(sign a)|a|^|b|^|c| + (sign a)|a|^|c|^|b| + (sign b)|b|^|a|^|c| + (sign b)|b|^|c|^|a| + (sign c)|c|^|a|^b| + (sign c)|c|^|b|^|a||;

|°ΕΣ|_{j∈{1, 2, 3}} a_j = ^|°^^+|_{j∈{1, 2, 3}} a_j = ^|°^^+|{a_j | j ∈ {1, 2, 3}} = a₁ ^|°^^+| a₂ ^|°^^+| a₃

= |a₁°|a₁|^|a₂|^|a₃| + a₁°|a₁|^|a₃|^|a₂| + a₂°|a₂|^|a₁|^|a₃| + a₂°|a₂|^|a₃|^|a₁| + a₃°|a₃|^|a₁|^|a₂| + a₃°|a₃|^|a₂|^|a₁||

= |(sign a₁)|a₁|^|a₂|^|a₃| + (sign a₁)|a₁|^|a₃|^|a₂| + (sign a₂)|a₂|^|a₁|^|a₃|

+ (sign a₂)|a₂|^|a₃|^|a₁| + (sign a₃)|a₃|^|a₁|^|a₂| + (sign a₃)|a₃|^|a₂|^|a₁||;

(-2) ^|°^^+| (-3) = |(-2)°|-2|^|-3| + (-3)°|-3|^|-2|| = |sign(-2)|-2|^|-3| + sign(-3)|-3|^|-2||

= |(-1)2^3 + (-1)3^2| = |(-1)2³ + (-1)3²| = |-17| = 17;

(-1) ^|°^^+| (-2) ^|°^^+| (-3)

= |(-1)°|-1|^|-2|^|-3| + (-1)°|-1|^|-3|^|-2| + (-2)°|-2|^|-1|^|-3| + (-2)°|-2|^|-3|^|-1| + (-3)°|-3|^|-1|^|-2| + (-3)°|-3|^|-2|^|-1||

= |(-1)|-1|^|-2|^|-3| + (-1)|-1|^|-3|^|-2| + (-1)|-2|^|-1|^|-3| + (-1)|-2|^|-3|^|-1| + (-1)|-3|^|-1|^|-2| + (-1)|-3|^|-2|^|-1||

= |(-1)(1^2^3 + 1^3^2 + 2^1^3 + 2^3^1 + 3^1^2 + 3^2^1)|

= |(-1)(1^8 + 1^9 + 2^1 + 2^3 + 3^1 + 3^2)| = |(-1)(1⁸ + 1⁹ + 2¹ + 2³ + 3¹ + 3²)| = |-24| = 24;

^|°^^+|2a = a ^|°^^+| a

= a₁ ^|°^^+| a₂ = |a₁°|a₁|^|a₂| + a₂°|a₂|^|a₁|| = |(sign a₁)|a₁|^|a₂| + (sign a₂)|a₂|^|a₁||

= |a°|a|^|a| + a°|a|^|a|| = |a°|a|^|a| + a°|a|^|a|| = |2a°|a|^|a|| = 2|a|^|a| ;

^|°^^+|3a = a ^|°^^+| a ^|°^^+| a = a₁ ^|°^^+| a₂ ^|°^^+| a₃

= |a₁°|a₁|^|a₂|^|a₃| + a₁°|a₁|^|a₃|^|a₂| + a₂°|a₂|^|a₁|^|a₃| + a₂°|a₂|^|a₃|^|a₁| + a₃°|a₃|^|a₁|^|a₂| + a₃°|a₃|^|a₂|^|a₁||

= |(sign a₁)|a₁|^|a₂|^|a₃| + (sign a₁)|a₁|^|a₃|^|a₂| + (sign a₂)|a₂|^|a₁|^|a₃|

+ (sign a₂)|a₂|^|a₃|^|a₁| + (sign a₃)|a₃|^|a₁|^|a₂| + (sign a₃)|a₃|^|a₂|^|a₁||

= |a°|a|^|a|^|a| + a°|a|^|a|^|a| + a°|a|^|a|^|a| + a°|a|^|a|^|a| + a°|a|^|a|^|a| + a°|a|^|a|^|a||

= |(sign a)|a|^|a|^|a| + (sign a)|a|^|a|^|a| + (sign a)|a|^|a|^|a| + (sign a)|a|^|a|^|a| + (sign a)|a|^|a|^|a| + (sign a)|a|^|a|^|a||

= |6a°|a|^|a|^|a|| = |6(sign a)|a|^|a|^|a|| = |6a°³|a|| = 6³|a|;

°^^+|2(-2) = (-2) ^|°^^+| (-2) = (-2)₁ ^|°^^+| (-2)₂

= |(-2)₁°|(-2)₁|^|(-2)₂| + (-2)₂°|(-2)₂|^|(-2)₁|| = |sign(-2)₁|(-2)₁|^|(-2)₂| + sign(-2)₂|(-2)₂|^|(-2)₁||

= |(-2)°|-2|^|-2| + (-2)°|-2|^|-2|| = |sign(-2)|-2|^|-2| + sign(-2)|-2|^|-2|| = |(-1)2² + (-1)2²| = |-8| = 8;

^|°^^+|3(-2) = (-2) ^|°^^+| (-2) ^|°^^+| (-2) = (-2)₁ ^|°^^+| (-2)₂ ^|°^^+| (-2)₃

= |(-2)₁°|(-2)₁|^|(-2)₂|^|(-2)₃| + (-2)₁°|(-2)₁|^|(-2)₃|^|(-2)₂| + (-2)₂°|(-2)₂|^|(-2)₁|^|(-2)₃|

+ (-2)₂°|(-2)₂|^|(-2)₃|^|(-2)₁| + (-2)₃°|(-2)₃|^|(-2)₁|^|(-2)₂| + (-2)₃°|(-2)₃|^|(-2)₂|^|(-2)₁||

= |(-2)°|-2|^|-2|^|-2| + (-2)°|-2|^|-2|^|-2| + (-2)°|-2|^|-2|^|-2| + (-2)°|-2|^|-2|^|-2| + (-2)°|-2|^|-2|^|-2| + (-2)°|-2|^|-2|^|-2||

= |6(-2)°|-2|^|-2|^|-2|| = |6(-1)|-2|^|-2|^|-2|| = |6(-1)³|-2|| = |6(-1)³2| = |6(-1)16| = |-96| = 96.

Commutative Power-Sum-Modulus-Sign Exponentiation

Idea

The principal idea of commutative power-sum-modulus-sign exponentiation is as follows:

consider any set of real numbers;

if there are coinciding (equal) numbers among them, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) numbers to artificially differentiate between them;

consider all the permutations of these already distinguishable numbers;

for every permutation of these numbers, compose their power tower using them in the order corresponding to this permutation;

for all these permutations together, compose the modulus (absolute value) of the sum of all these power towers;

multiply this modulus with the sign of the sum of all the initial (given) numbers;

in this expression, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative power-sum-modulus-sign exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative power-sum-modulus-sign exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative power-sum-modulus-sign exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative power-sum-modulus-sign exponential is denoted by

... ^|^^+|° a ^|^^+|° ... ^|^^+|° b ^|^^+|° ... ^|^^+|° c ^|^^+|° ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative power-sum-modulus-sign exponential is denoted by

|ΕΣ|°_j∈J a_j = ^|^^+|°_j∈J a_j = ^|^^+|°{a_j | j ∈ J}

= ... ^|^^+|° a_j' ^|^^+|° ... ^|^^+|° a_j'' ^|^^+|° ... ^|^^+|° a_j''' ^|^^+|° ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative power-sum-modulus-sign tetration is denoted by

^|^^+|°ⁿa = a ^|^^+|° a ^|^^+|° ... ^|^^+|° a

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a ^|^^+|° b = (a + b)°|a^b + b^a| = sign(a + b)|a^b + b^a|

= (a + b)°(|a^b + b^a|) = sign(a + b)(|a^b + b^a|);

a ^|^^+|° b ^|^^+|° c = (a + b + c)°|a^b^c + a^c^b + b^a^c + b^c^a + c^a^b + c^b^a|

= sign(a + b + c)|a^b^c + a^c^b + b^a^c + b^c^a + c^a^b + c^b^a|;

°|ΕΣ|_{j∈{1, 2, 3}} a_j = ^|^^+|°_{j∈{1, 2, 3}} a_j = ^|^^+|°{a_j | j ∈ {1, 2, 3}} = a₁ ^|^^+|° a₂ ^|^^+|° a₃

= (a₁ + a₂ + a₃)°|a₁^a₂^a₃ + a₁^a₃^a₂ + a₂^a₁^a₃ + a₂^a₃^a₁ + a₃^a₁^a₂ + a₃^a₂^a₁|

= sign(a₁ + a₂ + a₃)|a₁^a₂^a₃ + a₁^a₃^a₂ + a₂^a₁^a₃ + a₂^a₃^a₁ + a₃^a₁^a₂ + a₃^a₂^a₁|;

(-2) ^|^^+|° (-3) = [(-2) + (-3)]°|(-2)^(-3) + (-3)^(-2)| = sign[(-2) + (-3)]|(-2)^(-3) + (-3)^(-2)|

= (-1)|-1/2^3 + 1/3^2| = (-1)|-1/2³ + 1/3²| = -1/72;

(-1) ^|^^+|° (-2) ^|^^+|° (-3)

= [(-1) + (-2) + (-3)]°|(-1)^(-2)^(-3) + (-1)^(-3)^(-2) + (-2)^(-1)^(-3) + (-2)^(-3)^(-1) + (-3)^(-1)^(-2) + (-3)^(-2)^(-1)|

= sign[(-1) + (-2) + (-3)]|(-1)^(-2)^(-3) + (-1)^(-3)^(-2) + (-2)^(-1)^(-3) + (-2)^(-3)^(-1) + (-3)^(-1)^(-2) + (-3)^(-2)^(-1)|

does not exist when using the real numbers only because

(-1)^(-2)^(-3) = (-1)^(-1/2³) = (-1)^(-1/8) = 1/(-1)^(1/8),

(-3)^(-2)^(-1) = (-3)^(-1/2) = 1/(-3)^(1/2);

using base-sign-conserving exponentiation denoted by "^ , we obtain

(-1) ^|"^^+|° (-2) ^|"^^+|° (-3)

= [(-1) + (-2) + (-3)]°|(-1)"^(-2)"^(-3) + (-1)"^(-3)"^(-2) + (-2)"^(-1)"^(-3) + (-2)"^(-3)"^(-1) + (-3)"^(-1)"^(-2) + (-3)"^(-2)"^(-1)|

= (-1)|(-1)"^[-2^(-3)] + (-1)"^[-3^(-2)] + (-2)"^[-1^(-3)] + (-2)"^[-3^(-1)] + (-3)"^[-1^(-2)] + (-3)"^[-2^(-1)]|

= (-1)|(-1)"^(-1/8) + (-1)"^(-1/9) + (-2)"^(-1) + (-2)"^(-1/3) + (-3)"^(-1) + (-3)"^(-1/2)|

= (-1)|[-1^(-1/8)] + [-1^(-1/9)] + [-2^(-1)] + [-2^(-1/3)] + [-3^(-1)] + [-3^(-1/2)]|

= (-1)|(-1)(1 + 1 + 1/2 + 1/2^1/3 + 1/3 + 1/3^1/2)| = - 17/6 - 1/2^1/3 - 1/3^1/2 ;

^|^^+|°²a = a ^|^^+|° a

= a₁ ^|^^+|° a₂ = (a₁ + a₂)°|a₁^a₂ + a₂^a₁| = sign(a₁ + a₂)|a₁^a₂ + a₂^a₁|

= (a + a)°|a^a + a^a| = a°|a^a + a^a| = 2a°|a^a|;

^|^^+|°³a = a ^|^^+|° a ^|^^+|° a = a₁ ^|^^+|° a₂ ^|^^+|° a₃

= (a₁ + a₂ + a₃)°|a₁^a₂^a₃ + a₁^a₃^a₂ + a₂^a₁^a₃ + a₂^a₃^a₁ + a₃^a₁^a₂ + a₃^a₂^a₁|

= sign(a₁ + a₂ + a₃)|a₁^a₂^a₃ + a₁^a₃^a₂ + a₂^a₁^a₃ + a₂^a₃^a₁ + a₃^a₁^a₂ + a₃^a₂^a₁|

= (a + a + a)°|a^a^a + a^a^a + a^a^a + a^a^a + a^a^a + a^a^a|

= sign(a + a + a)|a^a^a + a^a^a + a^a^a + a^a^a + a^a^a + a^a^a|

= 6a°|a^a^a| = 6(sign a)|a^a^a| = 6a°|³a|;

^|^^+|°²(-2) = (-2) ^|^^+|° (-2) = (-2)₁ ^|^^+|° (-2)₂

= [(-2)₁ + (-2)₂]°|(-2)₁^(-2)₂ + (-2)₂^(-2)₁| = sign[(-2)₁ + (-2)₂]|(-2)₁^(-2)₂ + (-2)₂^(-2)₁|

= [(-2) + (-2)]°|(-2)^(-2) + (-2)^(-2)| = sign[(-2) + (-2)]|(-2)^(-2) + (-2)^(-2)| = (-1)(1/2² + 1/2²) = -1/2;

^|^^|+°³(-2) = (-2) ^|^^+|° (-2) ^|^^+|° (-2) = (-2)₁ ^|^^+|° (-2)₂ ^|^^+|° (-2)₃

= [(-2)₁ + (-2)₂ + (-2)₃]°|(-2)₁^(-2)₂^(-2)₃ + (-2)₁^(-2)₃^(-2)₂ + (-2)₂^(-2)₁^(-2)₃

+ (-2)₂^(-2)₃^(-2)₁ + (-2)₃^(-2)₁^(-2)₂ + (-2)₃^(-2)₂^(-2)₁|

= [(-2) + (-2) + (-2)]°|(-2)^(-2)^(-2) + (-2)^(-2)^(-2) + (-2)^(-2)^(-2) + (-2)^(-2)^(-2) + (-2)^(-2)^(-2) + (-2)^(-2)^(-2)|

= 6(-6)°|(-2)^(-2)^(-2)| = 6(-1)|(-2)^(-2)^(-2)| = 6(-1)|³(-2)| = 6(-1)|(-2)^(1/4)| = 6(-1)|(-2)^1/4|

does not exist due to (-2)^1/4 when using the real numbers only;

using the complex numbers, we obtain

^|^^+|°³(-2) = - 6 2^1/4 ;

using base-sign-conserving exponentiation denoted by "^ , we obtain

°^|"^^|+3(-2) = (-2) ^|"^^|+° (-2) ^|"^^|+° (-2) = (-2)₁ ^|"^^|+° (-2)₂ ^|"^^|+° (-2)₃

= [(-2)₁ + (-2)₂ + (-2)₃]°|(-2)₁"^(-2)₂"^(-2)₃ + (-2)₁"^(-2)₃"^(-2)₂ + (-2)₂"^(-2)₁"^(-2)₃

+ (-2)₂"^(-2)₃"^(-2)₁ + (-2)₃"^(-2)₁"^(-2)₂ + (-2)₃"^(-2)₂"^(-2)₁|

= [(-2) + (-2) + (-2)]°|(-2)"^(-2)"^(-2) + (-2)"^(-2)"^(-2) + (-2)"^(-2)"^(-2) + (-2)"^(-2)"^(-2) + (-2)"^(-2)"^(-2) + (-2)"^(-2)"^(-2)|

= 6(-6)°|(-2)"^(-2)"^(-2)| = 6(-6)°|³"(-2)| = 6(-6)°|(-2)"^[-2^(-2)]|

= 6(-6)°|(-2)"^(-2^-2)| = 6(-6)°|(-2)"^(-1/4)| = 6(-6)°|-2^(-1/4)| = 6(-1)|-2^-1/4| = - 6/2^1/4 .

Commutative Power-Modulus-Sum-Sign Exponentiation

Idea

The principal idea of commutative power-modulus-sum-sign exponentiation is as follows:

consider any set of real numbers;

if there are coinciding (equal) numbers among them, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) numbers to artificially differentiate between them;

consider all the permutations of these already distinguishable numbers;

for every permutation of these numbers, compose their power tower using them in the order corresponding to this permutation;

for all these permutations together, compose the sum of the moduli (absolute values) of all these power towers;

multiply this sum with the sign of the sum of all the initial (given) numbers;

in this expression, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative power-modulus-sum-sign exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative power-modulus-sum-sign exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative power-modulus-sum-sign exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative power-modulus-sum-sign exponential is denoted by

... ^|^^|+° a ^|^^|+° ... ^|^^|+° b ^|^^|+° ... ^|^^|+° c ^|^^|+° ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative power-modulus-sum-sign exponential is denoted by

|Ε|Σ°_j∈J a_j = ^|^^|+° _j∈J a_j = ^|^^|+°{a_j | j ∈ J}

= ... ^|^^|+° a_j' ^|^^|+° ... ^|^^|+° a_j'' ^|^^|+° ... ^|^^|+° a_j''' ^|^^|+° ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative power-modulus-sum-sign tetration is denoted by

^|^^|+°ⁿa = a ^|^^|+° a ^|^^|+° ... ^|^^|+° a

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a ^|^^|+° b = (a + b)°(|a^b| + |b^a|) = sign(a + b)(|a^b| + |b^a|)

= (a + b)°(|a^b| + |b^a|) = sign(a + b)(|a^b| + |b^a|);

a ^|^^|+° b ^|^^|+° c = (a + b + c)°(|a^b^c| + |a^c^b| + |b^a^c| + |b^c^a| + |c^a^b| + |c^b^a|)

= sign(a + b + c)(|a^b^c| + |a^c^b| + |b^a^c| + |b^c^a| + |c^a^b| + |c^b^a|);

°|Ε|Σ_{j∈{1, 2, 3}} a_j = ^|^^|+°_{j∈{1, 2, 3}} a_j = ^|^^|+°{a_j | j ∈ {1, 2, 3}} = a₁ ^|^^|+° a₂ ^|^^|+° a₃

= (a₁ + a₂ + a₃)°(|a₁^a₂^a₃| + |a₁^a₃^a₂| + |a₂^a₁^a₃| + |a₂^a₃^a₁| + |a₃^a₁^a₂| + |a₃^a₂^a₁|)

= sign(a₁ + a₂ + a₃)(|a₁^a₂^a₃| + |a₁^a₃^a₂| + |a₂^a₁^a₃| + |a₂^a₃^a₁| + |a₃^a₁^a₂| + |a₃^a₂^a₁|);

(-2) ^|^^|+° (-3) = [(-2) + (-3)]°(|(-2)^(-3)| + |(-3)^(-2)|) = sign[(-2) + (-3)](|(-2)^(-3)| + |(-3)^(-2)|)

= (-1)(|-1/2^3| + |-1/3^2|) = (-1)(1/2³ + 1/3²) = -17/72;

(-1) ^|^^|+° (-2) ^|^^|+° (-3)

= [(-1) + (-2) + (-3)]°[|(-1)^(-2)^(-3)| + |(-1)^(-3)^(-2)| + |(-2)^(-1)^(-3)| + |(-2)^(-3)^(-1)| + |(-3)^(-1)^(-2)| + |(-3)^(-2)^(-1)|]

= sign[(-1) + (-2) + (-3)][|(-1)^(-2)^(-3)| + |(-1)^(-3)^(-2)| + |(-2)^(-1)^(-3)| + |(-2)^(-3)^(-1)| + |(-3)^(-1)^(-2)| + |(-3)^(-2)^(-1)|]

does not exist when using the real numbers only because

|(-1)^(-2)^(-3)| = |(-1)^(-1/2³)| = |(-1)^(-1/8)| = |1/(-1)^(1/8)|,

|(-3)^(-2)^(-1)| = |(-3)^(-1/2)| = |1/(-3)^(1/2)|;

using the complex numbers, we obtain

(-1) ^|^^|+° (-2) ^|^^|+° (-3)

= [(-1) + (-2) + (-3)]°[|(-1)^(-2)^(-3)| + |(-1)^(-3)^(-2)| + |(-2)^(-1)^(-3)| + |(-2)^(-3)^(-1)| + |(-3)^(-1)^(-2)| + |(-3)^(-2)^(-1)]

= (-1)[|(-1)^(-1/8)| + |(-1)^(1/9)| + |(-2)^(-1)| + |(-2)^(-1/3)| + |(-3)^1| + |(-3)^(-1/2)|]

= (-1)(1 + 1 + 1/2 + 1/2^1/3 + 3 + 1/3^1/2) = - 11/2 - 1/2^1/3 - 1/3^1/2 ;

using base-sign-conserving exponentiation denoted by "^ , we obtain

(-1) ^|"^^|+° (-2) ^|"^^|+° (-3)

= [(-1) + (-2) + (-3)]°[|(-1)"^(-2)"^(-3)| + |(-1)"^(-3)"^(-2)| + |(-2)"^(-1)"^(-3)| + |(-2)"^(-3)"^(-1)| + |(-3)"^(-1)"^(-2)| + |(-3)"^(-2)"^(-1)|]

= (-1){|(-1)"^[-2^(-3)]| + |(-1)"^[-3^(-2)]| + |(-2)"^[-1^(-3)]| + |(-2)"^[-3^(-1)]| + |(-3)"^[-1^(-2)]| + |(-3)"^[-2^(-1)]|}

= (-1)[|(-1)"^(-1/8)| + |(-1)"^(-1/9)| + |(-2)"^(-1)| + |(-2)"^(-1/3)| + |(-3)"^(-1)| + |(-3)"^(-1/2)|]

= (-1)[|-1^(-1/8)| + |-1^(-1/9)| + |-2^(-1)| + |-2^(-1/3)| + |-3^(-1)| + |-3^(-1/2)|]

= (-1)(1 + 1 + 1/2 + 1/2^1/3 + 1/3 + 1/3^1/2) = - 17/6 - 1/2^1/3 - 1/3^1/2 ;

^|^^|+°²a = a ^|^^|+° a

= a₁ ^|^^|+° a₂ = (a₁ + a₂)°(|a₁^a₂| + |a₂^a₁|) = sign(a₁ + a₂)(|a₁^a₂| + |a₂^a₁|)

= (a + a)°(|a^a| + |a^a|) = a°(|a^a| + |a^a|) = 2a°|a^a|;

°^|^^|+3a = a ^|^^|+° a ^|^^|+° a = a₁ ^|^^|+° a₂ ^|^^|+° a₃

= (a₁ + a₂ + a₃)°[|a₁^a₂^a₃| + |a₁^a₃^a₂| + |a₂^a₁^a₃| + |a₂^a₃^a₁| + |a₃^a₁^a₂| + |a₃^a₂^a₁|]

= sign(a₁ + a₂ + a₃)(|a₁^a₂^a₃| + |a₁^a₃^a₂| + |a₂^a₁^a₃| + |a₂^a₃^a₁| + |a₃^a₁^a₂| + |a₃^a₂^a₁|)

= (a + a + a)°(|a^a^a| + |a^a^a| + |a^a^a| + |a^a^a| + |a^a^a| + |a^a^a|)

= sign(a + a + a)(|a^a^a| + |a^a^a| + |a^a^a| + |a^a^a| + |a^a^a| + |a^a^a|)

= 6a°|a^a^a| = 6(sign a)|a^a^a| = 6a°|³a|;

°^|^^|+2(-2) = (-2) ^|^^|+° (-2) = (-2)₁ ^|^^|+° (-2)₂

= [(-2)₁ + (-2)₂]°[|(-2)₁^(-2)₂| + |(-2)₂^(-2)₁|] = sign[(-2)₁ + (-2)₂][|(-2)₁^(-2)₂| + |(-2)₂^(-2)₁|]

= [(-2) + (-2)]°[|(-2)^(-2)| + |(-2)^(-2)|] = sign[(-2) + (-2)][|(-2)^(-2)| + |(-2)^(-2)|] = (-1)(1/2² + 1/2²) = -1/2;

^|^^|+°³(-2) = (-2) ^|^^|+° (-2) ^|^^|+° (-2) = (-2)₁ ^|^^|+° (-2)₂ ^|^^|+° (-2)₃

= [(-2)₁ + (-2)₂ + (-2)₃]°[|(-2)₁^(-2)₂^(-2)₃| + |(-2)₁^(-2)₃^(-2)₂| + |(-2)₂^(-2)₁^(-2)₃|

+ |(-2)₂^(-2)₃^(-2)₁| + |(-2)₃^(-2)₁^(-2)₂| + |(-2)₃^(-2)₂^(-2)₁|]

= [(-2) + (-2) + (-2)]°[|(-2)^(-2)^(-2)| + |(-2)^(-2)^(-2)| + |(-2)^(-2)^(-2)| + |(-2)^(-2)^(-2)| + |(-2)^(-2)^(-2)| + |(-2)^(-2)^(-2)|]

= 6(-2)°|(-2)^(-2)^(-2)| = 6(-1)|(-2)^(-2)^(-2)| = 6(-1)|³(-2)| = 6(-1)|(-2)^(1/4)| = 6(-1)|(-2)^1/4|

does not exist due to (-2)^1/4 when using the real numbers only;

using the complex numbers, we obtain

^|^^|+°³(-2) = - 6 2^1/4 ;

using base-sign-conserving exponentiation denoted by "^ , we obtain

°^|"^^|+3(-2) = (-2) ^|"^^|+° (-2) ^|"^^|+° (-2) = (-2)₁ ^|"^^|+° (-2)₂ ^|"^^|+° (-2)₃

= [(-2)₁ + (-2)₂ + (-2)₃]°[|(-2)₁"^(-2)₂"^(-2)₃| + |(-2)₁"^(-2)₃"^(-2)₂| + |(-2)₂"^(-2)₁"^(-2)₃|

+ |(-2)₂"^(-2)₃"^(-2)₁| + |(-2)₃"^(-2)₁"^(-2)₂| + |(-2)₃"^(-2)₂"^(-2)₁|]

= [(-2) + (-2) + (-2)]°[|(-2)"^(-2)"^(-2)| + |(-2)"^(-2)"^(-2)| + |(-2)"^(-2)"^(-2)| + |(-2)"^(-2)"^(-2)| + |(-2)"^(-2)"^(-2)| + |(-2)"^(-2)"^(-2)|]

= 6(-6)°|(-2)"^(-2)"^(-2)| = 6(-2)°|³"(-2)| = 6(-2)°|(-2)"^[-2^(-2)]|

= 6(-6)°|(-2)"^(-2^-2)| = 6(-6)°|(-2)"^(-1/4)| = 6(-6)°|-2^(-1/4)| = 6(-1)|-2^-1/4| = - 6/2^1/4 .

Commutative Modulus-Power-Sum-Sign Exponentiation

Idea

The principal idea of commutative modulus-power-sum-sign exponentiation is as follows:

for any set of real numbers, determine their moduli (absolute values);

if there are coinciding (equal) values among these moduli, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) values to artificially differentiate between them;

consider all the permutations of these already distinguishable values of all those moduli;

for every permutation of these moduli, compose their power tower using them in the order corresponding to this permutation;

for all these permutations together, compose the sum of all these power towers;

assign (attach) this sum the sign of the sum of the initial (given) numbers;

in the obtained expression, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative modulus-power-sum-sign exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative modulus-power-sum-sign exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative modulus-power-sum-sign exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative modulus-power-sum-sign exponential is denoted by

... ^||^⁺° a ^||^⁺° ... ^||^⁺° b ^||^⁺° ... ^||^⁺° c ^||^⁺° ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative modulus-power-sum-sign exponential is denoted by

||ΕΣ°_j∈J a_j = ^||^⁺°_j∈J a_j = ^||^⁺°{a_j | j ∈ J}

= ... ^||^⁺° a_j' ^||^⁺° ... ^||^⁺° a_j'' ^||^⁺° ... ^||^⁺° a_j''' ^||^⁺° ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative modulus-power-sum-sign tetration is denoted by

^||^⁺°ⁿa = a ^||^⁺° a ^||^⁺° ... ^||^⁺° a

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a ^||^⁺° b = (a + b)°(|a|^|b| + |b|^|a|) = sign(a + b)(|a|^|b| + |b|^|a|)

= (a + b)°(|a|^|b| + |b|^|a|) = sign(a + b)(|a|^|b| + |b|^|a|);

a ^||^⁺° b ^||^⁺° c = (a + b + c)°(|a|^|b|^|c| + |a|^|c|^|b| + |b|^|a|^|c| + |b|^|c|^|a| + |c|^|a|^|b| + |c|^|b|^|a|)

= sign(a + b + c)(|a|^|b|^|c| + |a|^|c|^|b| + |b|^|a|^|c| + |b|^|c|^|a| + |c|^|a|^|b| + |c|^|b|^|a|);

||ΕΣ°_{j∈{1, 2, 3}} a_j = ^||^⁺°_{j∈{1, 2, 3}} a_j = ^||^⁺°{a_j | j ∈ {1, 2, 3}} = a₁ ^||^⁺° a₂ ^||^⁺° a₃

= (a₁ + a₂ + a₃)°(|a₁|^|a₂|^|a₃| + |a₁|^|a₃|^|a₂| + |a₂|^|a₁|^|a₃| + |a₂|^|a₃|^|a₁| + |a₃|^|a₁|^|a₂| + |a₃|^|a₂|^|a₁|)

= sign(a₁ + a₂ + a₃)(|a₁|^|a₂|^|a₃| + |a₁|^|a₃|^|a₂| + |a₂|^|a₁|^|a₃| + |a₂|^|a₃|^|a₁| + |a₃|^|a₁|^|a₂| + |a₃|^|a₂|^|a₁|);

(-2) ^||^⁺° (-3) = [(-2) + (-3)]°(|-2|^|-3| + |-3|^|-2|) = sign[(-2) + (-3)](|-2|^|-3| + |-3|^|-2|)

= (-1)(2^3 + 3^2) = (-1)(2³ + 3²) = -17;

(-1) ^||^⁺° (-2) ^||^⁺° (-3)

= [(-1) + (-2) + (-3)]°(|-1|^|-2|^|-3| + |-1|^|-3|^|-2| + |-2|^|-1|^|-3| + |-2|^|-3|^|-1| + |-3|^|-1|^|-2| + |-3|^|-2|^|-1|)

= (-1)(|-1|^|-2|^|-3| + |-1|^|-3|^|-2| + |-2|^|-1|^|-3| + |-2|^|-3|^|-1| + |-3|^|-1|^|-2| + |-3|^|-2|^|-1|)

= (-1)(1^2^3 + 1^3^2 + 2^1^3 + 2^3^1 + 3^1^2 + 3^2^1)

= (-1)(1^8 + 1^9 + 2^1 + 2^3 + 3^1 + 3^2) = (-1)(1⁸ + 1⁹ + 2¹ + 2³ + 3¹ + 3²) = -24;

^||^⁺°²a = a ^||^⁺° a

= a₁ ^||^⁺° a₂ = (a₁ + a₂)°(|a₁|^|a₂| + |a₂|^|a₁|) = sign(a₁ + a₂)(|a₁|^|a₂| + |a₂|^|a₁|)

= (a + a)°(|a|^|a| + |a|^|a|) = a°(|a|^|a| + |a|^|a|) = 2a°|a|^|a| ;

°^||^⁺³a = a ^||^⁺° a ^||^⁺° a = a₁ ^||^⁺° a₂ ^||^⁺° a₃

= (a₁ + a₂ + a₃)°(|a₁|^|a₂|^|a₃| + |a₁|^|a₃|^|a₂| + |a₂|^|a₁|^|a₃| + |a₂|^|a₃|^|a₁| + |a₃|^|a₁|^|a₂| + |a₃|^|a₂|^|a₁|)

= sign(a₁ + a₂ + a₃)(|a₁|^|a₂|^|a₃| + |a₁|^|a₃|^|a₂| + |a₂|^|a₁|^|a₃| + |a₂|^|a₃|^|a₁| + |a₃|^|a₁|^|a₂| + |a₃|^|a₂|^|a₁|)

= (a + a + a)°(|a|^|a|^|a| + |a|^|a|^|a| + |a|^|a|^|a| + |a|^|a|^|a| + |a|^|a|^|a| + |a|^|a|^|a|)

= sign(a + a + a)(|a|^|a|^|a| + |a|^|a|^|a| + |a|^|a|^|a| + |a|^|a|^|a| + |a|^|a|^|a| + |a|^|a|^|a|)

= 6(3a)°|a|^|a|^|a| = 6(sign 3a)|a|^|a|^|a| = 6a°³|a|;

^||^⁺°²(-2) = (-2) ^||^⁺° (-2) = (-2)₁ ^||^⁺° (-2)₂

= [(-2)₁ + (-2)₂]°[|(-2)₁|^|(-2)₂| + |(-2)₂|^|(-2)₁|] = sign[(-2)₁ + (-2)₂][|(-2)₁|^|(-2)₂| + |(-2)₂|^|(-2)₁|]

= [(-2) + (-2)]°[|-2|^|-2| + |-2|^|-2|] = sign[(-2) + (-2)][|-2|^|-2| + |-2|^|-2|] = (-1)(2² + 2²) = -8;

^||^⁺°³(-2) = (-2) ^||^⁺° (-2) ^||^⁺° (-2) = (-2)₁ ^||^⁺° (-2)₂ ^||^⁺° (-2)₃

= [(-2)₁ + (-2)₂ + (-2)₃]°[|(-2)₁|^|(-2)₂|^|(-2)₃| + |(-2)₁|^|(-2)₃|^|(-2)₂| + |(-2)₂|^|(-2)₁|^|(-2)₃|

+ |(-2)₂|^|(-2)₃|^|(-2)₁| + |(-2)₃|^|(-2)₁|^|(-2)₂| + |(-2)₃|^|(-2)₂|^|(-2)₁|]

= [(-2) + (-2) + (-2)]°[|-2|^|-2|^|-2| + |-2|^|-2|^|-2| + |-2|^|-2|^|-2| + |-2|^|-2|^|-2| + |-2|^|-2|^|-2| + |-2|^|-2|^|-2|]

= sign[(-2) + (-2) + (-2)][|-2|^|-2|^|-2| + |-2|^|-2|^|-2| + |-2|^|-2|^|-2| + |-2|^|-2|^|-2| + |-2|^|-2|^|-2| + |-2|^|-2|^|-2|]

= 6(-6)°|-2|^|-2|^|-2| = 6(-1)|-2|^|-2|^|-2| = 6(-1)³|-2| = 6(-1)³2 = 6(-1)16 = -96.

Commutative Power-Sum Maximum-Exponentiation Type

Commutative Power-Sum Maximum-Exponentiation Mode

Commutative Power-Sum Maximum-Exponentiation

Idea

The principal idea of commutative power-sum maximum-exponentiation is as follows:

consider any set of real numbers;

if there are coinciding (equal) numbers, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) numbers to artificially differentiate between them;

consider all the permutations of these already distinguishable numbers;

for every permutation of these numbers, compose their power tower using them in the order corresponding to this permutation and replacing each exponent smaller than 1 with its multiplicative inverse (reciprocal);

for all these permutations together, compose the sum of all these power towers;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative power-sum maximum-exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative power-sum maximum-exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative power-sum maximum-exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative power-sum maximum-exponential is denoted by

... ^?^⁺ a ^?^⁺ ... ^?^⁺ b ^?^⁺ ... ^?^⁺ c ^?^⁺ ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative power-sum maximum-exponential is denoted by

Ε^?Σ_j∈J a_j = ^?^⁺_j∈J a_j = Ε^?Σ{a_j | j ∈ J} = ^?^⁺{a_j | j ∈ J}

= ... ^?^⁺ a_j' ^?^⁺ ... ^?^⁺ a_j'' ^?^⁺ ... ^?^⁺ a_j''' ^?^⁺ ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative power-sum maximum-tetration is denoted by

^?^⁺ⁿa = a ^?^⁺ a ^?^⁺ ... ^?^⁺ a

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a ^?^⁺ b = a^{max(b , 1/b)} + b^{max(a , 1/a)} = a^max(b , 1/b) + b^max(a, 1/a);

a ^?^⁺ b ^?^⁺ c = a^max(b , 1/b)^max(c , 1/c) + a^max(c , 1/c)^max(b , 1/b) + b^max(a , 1/a)^max(c , 1/c)

+ b^max(c , 1/c)^max(a , 1/a) + c^max(a , 1/a)^max(b , 1/b) + c^max(b , 1/b)^max(a , 1/a);

Ε^?Σ_{j∈{1, 2, 3}} a_j = Ε^?Σ{a_j | j ∈ {1, 2, 3}} = ^?^⁺_{j∈{1, 2, 3}} a_j = ^?^⁺{a_j | j ∈ {1, 2, 3}}

= a₁ ^?^⁺ a₂ ^?^⁺ a₃

= a₁^max(a₂ , 1/a₂)^max(a₃ , 1/a₃) + a₁^max(a₃ , 1/a₃)^max(a₂ , 1/a₂) + a₂^max(a₁ , 1/a₁)^max(a₃ , 1/a₃)

+ a₂^max(a₃ , 1/a₃)^max(a₁ , 1/a₁) + a₃^max(a₁ , 1/a₁)^max(a₂ , 1/a₂) + a₃^max(a₂ , 1/a₂)^max(a₁ , 1/a₁);

2 ^?^⁺ (1/3) = 2^max(1/3, 1/(1/3)) + (1/3)^max(2, 1/2) = 2^{max(1/3, 1/(1/3))} + (1/3)^{max(2, 1/2)}

= 2^{max(1/3, 3)} + (1/3)^{max(2, 1/2)} = 2³ + (1/3)² = 73/9;

1 ^?^⁺ (1/2) ^?^⁺ (1/3) = 1^max(1/2, 2)^max(1/3, 3) + 1^max(1/3, 3)^max(1/2, 2) + (1/2)^max(1, 1)^max(1/3, 3)

+ (1/2)^max(1/3, 3)^max(1, 1) + (1/3)^max(1, 1)^max(1/2, 2) + (1/3)^max(1/2, 2)^max(1, 1)

= 1^2^3 + 1^3^2 + (1/2)^1^3 + (1/2)^3^1 + (1/3)^1^2 + (1/3)^2^1

= 1^8 + 1^9 + (1/2)^1 + (1/2)^3 + (1/3)^1 + (1/3)^2

= 1⁸ + 1⁹ + 1/2¹ + 1/2³ + 1/3¹ + 1/3² = 1 + 1 + 1/2 + 1/8 + 1/3 + 1/9 = 221/72;

^?^⁺²a = a ^?^⁺ a = a₁ ^?^⁺ a₂

= a₁^max(a₂ , 1/a₂) + a₂^max(a₁ , 1/a₁)

= a^max(a , 1/a) + a^max(a , 1/a) = a^{max(a , 1/a)} + a^{max(a , 1/a)} = 2a^{max(a , 1/a)} ;

^?^⁺³a = a ^?^⁺ a ^?^⁺ a = a₁ ^?^⁺ a₂ ^?^⁺ a₃

= a₁^max(a₂ , 1/a₂)^max(a₃ , 1/a₃) + a₁^max(a₃ , 1/a₃)^max(a₂ , 1/a₂) + a₂^max(a₁ , 1/a₁)^max(a₃ , 1/a₃)

+ a₂^max(a₃ , 1/a₃)^max(a₁ , 1/a₁) + a₃^max(a₁ , 1/a₁)^max(a₂ , 1/a₂) + a₃^max(a₂ , 1/a₂)^max(a₁ , 1/a₁)

= a^max(a , 1/a)^max(a , 1/a) + a^max(a , 1/a)^max(a , 1/a) + a^max(a , 1/a)^max(a , 1/a)

+ a^max(a , 1/a)^max(a , 1/a) + a^max(a , 1/a)^max(a , 1/a) + a^max(a , 1/a)^max(a , 1/a)

= 6a^max(a , 1/a)^max(a , 1/a) = 6a^ ²max(a , 1/a);

(1/2) ^?^⁺ (1/2) = 2(1/2)^max(1/2, 2) = 2(1/2)² = 1/2;

(1/2) ^?^⁺ (1/2) ^?^⁺ (1/2) = (1/2)₁ ^?^⁺ (1/2)₂ ^?^⁺ (1/2)₃

= (1/2)₁^max[(1/2)₂|, 1/(1/2)₂]^max[(1/2)₃|, 1/(1/2)₃] + (1/2)₁^max[(1/2)₃ , 1/(1/2)₃]^max[(1/2)₂ , 1/(1/2)₂]

+ (1/2)₂^max[(1/2)₁|, 1/|(1/2)₁]^max[(1/2)₃ , 1/(1/2)₃] + (1/2)₂^max[(1/2)₃ , 1/|(1/2)₃]^max[(1/2)₁ , 1/|(1/2)₁]

+ (1/2)₃^max[(1/2)₁ , 1/(1/2)₁]^max[(1/2)₂ , 1/(1/2)₂] + |(1/2)₃^max[(1/2)₂ , 1/(1/2)₂]^max[(1/2)₁ , 1/|(1/2)₁]

= (1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)] + (1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]

+ (1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)] + (1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]

+ (1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)] + (1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]

= 6(1/2)^max[1/2, 1/(1/2)]^max[(1/2), 1/(1/2)] = 6(1/2)^max(1/2, 2)^max(1/2, 2)

= 6(1/2)^2^2 = 6(1/2)⁴ = 3/8.

Commutative Power-Modulus-Sum Maximum-Exponentiation

Idea

The principal idea of commutative power-modulus-sum maximum-exponentiation is as follows:

consider any set of real numbers;

if there are coinciding (equal) numbers, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) numbers to artificially differentiate between them;

consider all the permutations of these already distinguishable numbers;

for every permutation of these numbers, compose their power tower using them in the order corresponding to this permutation and replacing each exponent smaller than 1 with its multiplicative inverse (reciprocal);

for all these permutations together, compose the sum of the moduli (absolute values) of all these power towers;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative power-modulus-sum maximum-exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative power-modulus-sum maximum-exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative power-modulus-sum maximum-exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative power-modulus-sum maximum-exponential is denoted by

... ^?|^^|+ a ^?|^^|+ ... ^?|^^|+ b ^?|^^|+ ... ^?|^^|+ c ^?|^^|+ ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative power-modulus-sum maximum-exponential is denoted by

|Ε|^?Σ_j∈J a_j = ^?|^^|+_j∈J a_j = |Ε|^?Σ{a_j | j ∈ J} = ^?|^^|+{a_j | j ∈ J}

= ... ^?|^^|+ a_j' ^?|^^|+ ... ^?|^^|+ a_j'' ^?|^^|+ ... ^?|^^|+ a_j''' ^?|^^|+ ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative power-modulus-sum maximum-tetration is denoted by

^?|^^|+na = a ^?|^^|+ a ^?|^^|+ ... ^?|^^|+ a

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a ^?|^^|+ b = |a^max(b , 1/b)| + |b^max(a , 1/a)| = |a^{max(b , 1/b)}| + |b^{max(a , 1/a)}|;

a ^?|^^|+ b ^?|^^|+ c = |a^max(b , 1/b)^max(c , 1/c)| + |a^max(c , 1/c)^max(b , 1/b)| + |b^max(a , 1/a)^max(c , 1/c)|

+ |b^max(c , 1/c)^max(a , 1/a)| + |c^max(a , 1/a)^max(b , 1/b)| + |c^max(b , 1/b)^max(a , 1/a)|;

|Ε|^?Σ_{j∈{1, 2, 3}} a_j = |Ε|^?Σ{a_j | j ∈ {1, 2, 3}} = ^?|^^|+_{j∈{1, 2, 3}} a_j = ^?|^^|+{a_j | j ∈ {1, 2, 3}}

= a₁ ^?|^^|+ a₂ ^?|^^|+ a₃

= |a₁^max(a₂ , 1/a₂)^max(a₃ , 1/a₃)| + |a₁^max(a₃ , 1/a₃)^max(a₂ , 1/a₂)| + |a₂^max(a₁ , 1/a₁)^max(a₃ , 1/a₃)|

+ |a₂^max(a₃ , 1/a₃)^max(a₁ , 1/a₁)| + |a₃^max(a₁ , 1/a₁)^max(a₂ , 1/a₂)| + |a₃^max(a₂ , 1/a₂)^max(a₁ , 1/a₁)|;

(-2) ^?|^^|+ (1/3) = |(-2)^max[(1/3), 1/(1/3)]| + |(1/3)^max[(-2), 1/(-2)]| = |(-2)^{max[(1/3), 1/(1/3)]}| + |(1/3)^{max[(-2), 1/(-2)]}|

= |(-2)^max(1/3, 3)| + |(1/3)^max(-2, -1/2)| = |(-2)^{max(1/3, 3)}| + |(1/3)^{max(-2, -1/2)}|

= |(-2)^3| + |(1/3)^(-1/2)| = |(-2)³| + |(1/3)^-1/2| = |-8| + |3^1/2| = 8 + 3^1/2 ;

1 ^?|^^|+ (-1/3) ^?|^^|+ (-1/5) =

= |1^max[(-1/3), 1/(-1/3)]^max[(-1/5), 1/(-1/5)]| + |1^max[(-1/5), 1/(-1/5)]^max[(-1/3), 1/(-1/3)]|

+ |(-1/3)^max(1, 1/1)^max[(-1/5), 1/(-1/5)]| + |(-1/3)^max[(-1/5), 1/(-1/5)]^max(1, 1/1)|

+ |(-1/5)^max(1, 1/1)^max[(-1/3), 1/(-1/3)]| + |(-1/5)^max[(-1/3), 1/(-1/3)]^max(1, 1/1)|

= |1^max(-1/3, -3)^max(-1/5, -5)| + |1^max(-1/5, -5)^max(-1/3, -3)|

+ |(-1/3)^max(1, 1)^max(-1/5, -5)| + |(-1/3)^max(-1/5, -5)^max(1, 1)|

+ |(-1/5)^max(1, 1)^max(-1/3, -3)| + |(-1/5)^max(-1/3, -3)^max(1, 1)|

= |1^(-1/3)^(-1/5)| + |1^(-1/5)^(-1/3)|

+ |(-1/3)^1^(-1/5)| + |(-1/3)^(-1/5)^1|

+ |(-1/5)^1^(-1/3)| + |(-1/5)^(-1/3)^1|

= |1| + |1| + |-1/3| + |(-1/3)^(-1/5)| + |-1/5| + |(-1/5)^(-1/3)|

= 1 + 1 + 1/3 + |(-3)^(1/5)| + 1/5 + |(-5)^(1/3)|

= 38/15 + 3^1/5 + 5^1/3;

^?|^^|+2a = a ^?|^^|+ a = a₁ ^||?^⁺ a₂

= |a₁^max(a₂ , 1/a₂)| + |a₂^max(a₁ , 1/a₁)|

= |a^max(a , 1/a)| + |a^max(a , 1/a)| = |a^{max(|a|, 1/|a|)}| + |a|^{max(|a|, 1/|a|)} = 2|a|^{max(|a|, 1/|a|)} ;

^?|^^|+3a = a ^?|^^|+ a ^?|^^|+ a = a₁ ^?|^^|+ a₂ ^?|^^|+ a₃

= |a₁^max(a₂ , 1/a₂)^max(a₃ , 1/a₃)| + |a₁^max(a₃ , 1/a₃)^max(a₂ , 1/a₂)| + |a₂^max(a₁ , 1/a₁)^max(a₃ , 1/a₃)|

+ |a₂^max(a₃ , 1/a₃)^max(a₁ , 1/a₁)| + |a₃^max(a₁ , 1/a₁)^max(a₂ , 1/a₂)| + |a₃^max(a₂ , 1/a₂)^max(a₁ , 1/a₁)|

= |a^max(a , 1/a)^max(a , 1/a)| + |a^max(a , 1/a)^max(a , 1/a)| + |a^max(a , 1/a)^max(a , 1/a)|

+ |a^max(a , 1/a)^max(a , 1/a)| + |a^max(a , 1/a)^max(a , 1/a)| + |a^max(a , 1/a)^max(a , 1/a)|

= 6|a^max(a , 1/a)^max(a , 1/a)| = 6|a^ ²max(a , 1/a)|;

^?|^^|+2(-1/2) = (-1/2) ^?|^^|+ (-1/2)

= |(-1/2)₁^max[(-1/2)₂ , 1/(-1/2)₂]| + |(-1/2)₂^max[(-1/2)₁ , 1/(-1/2)₁]|

= |(-1/2)^max[(-1/2), 1/(-1/2)]| + |(-1/2)^max[(-1/2), 1/(-1/2)]|

= 2|(-1/2)^max[(-1/2), 1/(-1/2)]| = 2|(-1/2)^(-1/2)| = 2|(-2)^(1/2)| = 2|(-2)^1/2|

does not exist in the real numbers whereas in the complex numbers

^?|^^|+2(-1/2) = (-1/2) ^?|^^|+ (-1/2)

= 2|(-2)^1/2| = 2 2^1/2 = 2^3/2

and in negative base power theory,

^?|"^^|+2(-1/2) = (-1/2) ^?|"^^|+ (-1/2)

= 2|(-2)"^1/2| = 2|-2^1/2| = 2 2^1/2 = 2^3/2 ;

^?|^^|+3(1/2) = (1/2) ^?|^^|+ (1/2) ^?|^^|+ (1/2) = (1/2)₁ ^?|^^|+ (1/2)₂ ^?|^^|+ (1/2)₃

= |(1/2)₁^max[(1/2)₂|, 1/(1/2)₂]^max[(1/2)₃|, 1/(1/2)₃]| + |(1/2)₁^max[(1/2)₃ , 1/(1/2)₃]^max[(1/2)₂ , 1/(1/2)₂]|

+ |(1/2)₂^max[(1/2)₁|, 1/|(1/2)₁]^max[(1/2)₃ , 1/(1/2)₃]| + |(1/2)₂^max[(1/2)₃ , 1/|(1/2)₃]^max[(1/2)₁ , 1/|(1/2)₁]|

+ |(1/2)₃^max[(1/2)₁ , 1/(1/2)₁]^max[(1/2)₂ , 1/(1/2)₂]| + |(1/2)₃^max[(1/2)₂ , 1/(1/2)₂]^max[(1/2)₁ , 1/|(1/2)₁]|

= |(1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]| + |(1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]|

+ |(1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]| + |(1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]|

+ |(1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]| + |(1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]|

= 6|(1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]| = 6|(1/2)^max(1/2, 2)^max(1/2, 2)|

= 6|(1/2)^2^2| = 6|(1/2)⁴| = 6(1/2)⁴ = 3/8.

Commutative Power-Sum-Modulus Maximum-Exponentiation

Idea

The principal idea of commutative power-sum-modulus maximum-exponentiation is as follows:

consider any set of real numbers;

if there are coinciding (equal) numbers, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) numbers to artificially differentiate between them;

consider all the permutations of these already distinguishable numbers;

for every permutation of these numbers, compose their power tower using them in the order corresponding to this permutation and replacing each exponent smaller than 1 with its multiplicative inverse (reciprocal);

for all these permutations together, compose the modulus (absolute value) of the sum of all these power towers;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative power-sum-modulus maximum-exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative power-sum-modulus maximum-exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative power-sum-modulus maximum-exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative power-sum-modulus maximum-exponential is denoted by

... ^?^^|+| a ^?^^|+| ... ^?^^|+| b ^?^^|+| ... ^?^^|+| c ^?^^|+| ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative power-sum-modulus maximum-exponential is denoted by

Ε^?|Σ|_j∈J a_j = ^?^^|+|_j∈J a_j = Ε^?|Σ|{a_j | j ∈ J} = ^?^^|+|{a_j | j ∈ J}

= ... ^?^^|+| a_j' ^?^^|+| ... ^?^^|+| a_j'' ^?^^|+| ... ^?^^|+| a_j''' ^?^^|+| ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative power-sum-modulus maximum-tetration is denoted by

^?^^|+|na = a ^?^^|+| a ^?^^|+| ... ^?^^|+| a

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a ^?^^|+| b = |a^max(b , 1/b) + b^max(a , 1/a)| = |a^{max(b , 1/b)} + b^{max(a , 1/a)}|;

a ^?^^|+| b ^?^^|+| c = |a^max(b , 1/b)^max(c , 1/c) + a^max(c , 1/c)^max(b , 1/b) + b^max(a , 1/a)^max(c , 1/c)

+ b^max(c , 1/c)^max(a , 1/a) + c^max(a , 1/a)^max(b , 1/b) + c^max(b , 1/b)^max(a , 1/a)|;

Ε^?|Σ|_{j∈{1, 2, 3}} a_j = Ε^?|Σ|{a_j | j ∈ {1, 2, 3}} = ^?^^|+|_{j∈{1, 2, 3}} a_j = ^?^^|+|{a_j | j ∈ {1, 2, 3}}

= a₁ ^?^^|+| a₂ ^?^^|+| a₃

= |a₁^max(a₂ , 1/a₂)^max(a₃ , 1/a₃) + a₁^max(a₃ , 1/a₃)^max(a₂ , 1/a₂) + a₂^max(a₁ , 1/a₁)^max(a₃ , 1/a₃)

+ a₂^max(a₃ , 1/a₃)^max(a₁ , 1/a₁) + a₃^max(a₁ , 1/a₁)^max(a₂ , 1/a₂) + a₃^max(a₂ , 1/a₂)^max(a₁ , 1/a₁)|;

(-2) ^?^^|+| (1/3) = |(-2)^max[(1/3), 1/(1/3)] + (1/3)^max[(-2), 1/(-2)]| = |(-2)^{max[(1/3), 1/(1/3)]} + (1/3)^{max[(-2), 1/(-2)]}|

= |(-2)^max(1/3, 3) + (1/3)^max(-2, -1/2)| = |(-2)^{max(1/3, 3)} + (1/3)^{max(-2, -1/2)}|

= |(-2)^3 + (1/3)^(-1/2)| = |(-2)³ + (1/3)^-1/2| = |-8 + 3^1/2| = 8 - 3^1/2 ;

1 ^?^^|+| (-1/3) ^?^^|+| (-1/5) =

= |1^max[(-1/3), 1/(-1/3)]^max[(-1/5), 1/(-1/5)] + 1^max[(-1/5), 1/(-1/5)]^max[(-1/3), 1/(-1/3)]

+ (-1/3)^max(1, 1/1)^max[(-1/5), 1/(-1/5)] + (-1/3)^max[(-1/5), 1/(-1/5)]^max(1, 1/1)

+ (-1/5)^max(1, 1/1)^max[(-1/3), 1/(-1/3)] + (-1/5)^max[(-1/3), 1/(-1/3)]^max(1, 1/1)|

= |1^max(-1/3, -3)^max(-1/5, -5) + 1^max(-1/5, -5)^max(-1/3, -3)

+ (-1/3)^max(1, 1)^max(-1/5, -5) + (-1/3)^max(-1/5, -5)^max(1, 1)

+ (-1/5)^max(1, 1)^max(-1/3, -3) + (-1/5)^max(-1/3, -3)^max(1, 1)|

= |1^(-1/3)^(-1/5) + 1^(-1/5)^(-1/3)

+ (-1/3)^1^(-1/5) + (-1/3)^(-1/5)^1

+ (-1/5)^1^(-1/3) + (-1/5)^(-1/3)^1|

= |1 + 1 + (-1/3) + (-1/3)^(-1/5) + (-1/5) + (-1/5)^(-1/3)|

= 1 + 1 - 1/3 + (-3)^(1/5) - 1/5 + (-5)^(1/3)|

= 22/15 - 3^1/5 - 5^1/3 ;

^?^^|+|2a = a ^?^^|+| a = a₁ ^?^^|+| a₂

= |a₁^max(a₂ , 1/a₂) + a₂^max(a₁ , 1/a₁)|

= |a^max(a , 1/a) + a^max(a , 1/a)| = |a^{max(|a|, 1/|a|)} + a|^{max(|a|, 1/|a|)} = 2|a|^{max(|a|, 1/|a|)} ;

^?^^|+|3a = a ^?^^|+| a ^?^^|+| a = a₁ ^?^^|+| a₂ ^?^^|+| a₃

= |a₁^max(a₂ , 1/a₂)^max(a₃ , 1/a₃) + a₁^max(a₃ , 1/a₃)^max(a₂ , 1/a₂) + a₂^max(a₁ , 1/a₁)^max(a₃ , 1/a₃)

+ a₂^max(a₃ , 1/a₃)^max(a₁ , 1/a₁) + a₃^max(a₁ , 1/a₁)^max(a₂ , 1/a₂) + a₃^max(a₂ , 1/a₂)^max(a₁ , 1/a₁)|

= |a^max(a , 1/a)^max(a , 1/a) + a^max(a , 1/a)^max(a , 1/a) + a^max(a , 1/a)^max(a , 1/a)

+ a^max(a , 1/a)^max(a , 1/a) + a^max(a , 1/a)^max(a , 1/a) + a^max(a , 1/a)^max(a , 1/a)|

= 6|a^max(a , 1/a)^max(a , 1/a)| = 6|a^ ²max(a , 1/a)|;

^?^^|+|2(-1/2) = (-1/2) ^?^^|+| (-1/2)

= |(-1/2)₁^max[(-1/2)₂ , 1/(-1/2)₂] + (-1/2)₂^max[(-1/2)₁ , 1/(-1/2)₁]|

= |(-1/2)^max[(-1/2), 1/(-1/2)] + (-1/2)^max[(-1/2), 1/(-1/2)]|

= 2|(-1/2)^max[(-1/2), 1/(-1/2)]| = 2|(-1/2)^(-1/2)| = 2|(-2)^(1/2)| = 2|(-2)^1/2|

does not exist in the real numbers whereas in the complex numbers

^?^^|+|2(-1/2) = (-1/2) ^?^^|+| (-1/2)

= 2|(-2)^1/2| = 2 2^1/2 = 2^3/2

and in negative base power theory,

^?"^^|+|2(-1/2) = (-1/2) ^?"^^|+| (-1/2)

= 2|(-2)"^1/2| = 2|-2^1/2| = 2 2^1/2 = 2^3/2 ;

^?^^|+|3(1/2) = (1/2) ^?^^|+| (1/2) ^?^^|+| (1/2) = (1/2)₁ ^?^^|+| (1/2)₂ ^?^^|+| (1/2)₃

= |(1/2)₁^max[(1/2)₂|, 1/(1/2)₂]^max[(1/2)₃|, 1/(1/2)₃] + (1/2)₁^max[(1/2)₃ , 1/(1/2)₃]^max[(1/2)₂ , 1/(1/2)₂]

+ (1/2)₂^max[(1/2)₁|, 1/|(1/2)₁]^max[(1/2)₃ , 1/(1/2)₃] + (1/2)₂^max[(1/2)₃ , 1/|(1/2)₃]^max[(1/2)₁ , 1/|(1/2)₁]

+ (1/2)₃^max[(1/2)₁ , 1/(1/2)₁]^max[(1/2)₂ , 1/(1/2)₂] + (1/2)₃^max[(1/2)₂ , 1/(1/2)₂]^max[(1/2)₁ , 1/|(1/2)₁]|

= |(1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)] + (1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]

+ (1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)] + (1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]

+ (1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)] + (1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]|

= 6|(1/2)^max[1/2, 1/(1/2)]^max[1/2, 1/(1/2)]| = 6|(1/2)^max(1/2, 2)^max(1/2, 2)|

= 6|(1/2)^2^2| = 6|(1/2)⁴| = 6(1/2)⁴ = 3/8.

Commutative Modulus-Power-Sum Maximum-Exponentiation

Idea

The principal idea of commutative modulus-power-sum maximum-exponentiation is as follows:

for any set of real numbers, determine their moduli (absolute values);

if there are coinciding (equal) values among these moduli, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) values to artificially differentiate between them;

consider all the permutations of these already distinguishable values of all those moduli;

for every permutation of these values, compose their power tower using them in the order corresponding to this permutation and replacing each exponent smaller than 1 with its multiplicative inverse (reciprocal);

for all these permutations together, compose the sum of all these power towers;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative modulus-power-sum maximum-exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative modulus-power-sum maximum-exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative modulus-power-sum maximum-exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative modulus-power-sum maximum-exponential is denoted by

... ^||?^⁺ a ^||?^⁺ ... ^||?^⁺ b ^||?^⁺ ... ^||?^⁺ c ^||?^⁺ ... = ... ^?^⁺ |a| ^?^⁺ ... ^?^⁺ |b| ^?^⁺ ... ^?^⁺ |c| ^?^⁺ ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative modulus-power-sum maximum-exponential is denoted by

Ε^||?Σ_j∈J a_j = ^||?^⁺_j∈J a_j = Ε^||?Σ{a_j | j ∈ J} = ^||?^⁺{a_j | j ∈ J}

= Ε^?Σ_j∈J |a_j| = ^?^⁺_j∈J |a_j| = Ε^?Σ{|a_j| | j ∈ J} = ^?^⁺{|a_j| | j ∈ J}

= ... ^||?^⁺ a_j' ^||?^⁺ ... ^||?^⁺ a_j'' ^||?^⁺ ... ^||?^⁺ a_j''' ^||?^⁺ ...

= ... ^?^⁺ |a_j'| ^?^⁺ ... ^?^⁺ |a_j''| ^?^⁺ ... ^?^⁺ |a_j'''| ^?^⁺ ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative modulus-power-sum maximum-tetration is denoted by

^||?^⁺ⁿa = a ^||?^⁺ a ^||?^⁺ ... ^||?^⁺ a

= ^?^⁺ⁿ|a| = |a| ^?^⁺ |a| ^?^⁺ ... ^?^⁺ |a|

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a ^||?^⁺ b = |a| ^?^⁺ |b| = |a|^max(|b|, 1/|b|) + |b|^max(|a|, 1/|a|) = |a|^{max(|b|, 1/|b|)} + |b|^{max(|a|, 1/|a|)} ;

a ^||?^⁺ b ^||?^⁺ c = |a| ^?^⁺ |b| ^?^⁺ |c|

= |a|^max(|b|, 1/|b|)^max(|c|, 1/|c|) + |a|^max(|c|, 1/|c|)^max(|b|, 1/|b|) + |b|^max(|a|, 1/|a|)^max(|c|, 1/|c|)

+ |b|^max(|c|, 1/|c|)^max(|a|, 1/|a|) + |c|^max(|a|, 1/|a|)^max(|b|, 1/|b|) + |c|^max(|b|, 1/|b|)^max(|a|, 1/|a|);

Ε^||?Σ_{j∈{1, 2, 3}} a_j = Ε^||?Σ{a_j | j ∈ {1, 2, 3}} = ^||?^⁺_{j∈{1, 2, 3}} a_j = ^||?^⁺{a_j | j ∈ {1, 2, 3}}

= Ε^?Σ_{j∈{1, 2, 3}} |a_j| = Ε^?Σ{|a_j| | j ∈ {1, 2, 3}} = ^?^⁺_{j∈{1, 2, 3}} |a_j| = ^?^⁺{|a_j| | j ∈ {1, 2, 3}}

= a₁ ^||?^⁺ a₂ ^||?^⁺ a₃ = |a₁| ^?^⁺ |a₂| ^?^⁺ |a₃|

= |a₁|^max(|a₂|, 1/|a₂|)^max(|a₃|, 1/|a₃|) + |a₁|^max(|a₃|, 1/|a₃|)^max(|a₂|, 1/|a₂|) + |a₂|^max(|a₁|, 1/|a₁|)^max(|a₃|, 1/|a₃|)

+ |a₂|^max(|a₃|, 1/|a₃|)^max(|a₁|, 1/|a₁|) + |a₃|^max(|a₁|, 1/|a₁|)^max(|a₂|, 1/|a₂|) + |a₃|^max(|a₂|, 1/|a₂|)^max(|a₁|, 1/|a₁|);

(-2) ^||?^⁺ (-1/3) = |-2| ^?^⁺ |-1/3| = 2 ^?^⁺ (1/3)

= |-2|^max(|-1/3|, 1/|-1/3|) + |-1/3|^max(|-2|, 1/|-2|) = |-2|^{max(|-1/3|, 1/|-1/3|)} + |-1/3|^{max(|-2|, 1/|-2|)}

= 2^max(1/3, 1/(1/3)) + (1/3)^max(2, 1/2) = 2^{max(1/3, 1/(1/3))} + (1/3)^{max(2, 1/2)}

= 2^{max(1/3, 3)} + (1/3)^{max(2, 1/2)} = 2³ + (1/3)² = 73/9;

(-1) ^||?^⁺ (-1/2) ^||?^⁺ (-1/3) = |-1| ^?^⁺ |-1/2| ^?^⁺ |-1/3| = 1 ^?^⁺ (1/2) ^?^⁺ (1/3)

= |-1|^max(|-1/2|, 1/|-1/2|)^max(|-1/3|, 1/|-1/3|) + |-1|^max(|-1/3|, 1/|-1/3|)^max(|-1/2|, 1/|-1/2|) + |-1/2|^max(|-1|, 1/|-1|)^max(|-1/3|, 1/|-1/3|)

+ |-1/2|^max(|-1/3|, 1/|-1/3|)^max(|-1|, 1/|-1|) + |-1/3|^max(|-1|, 1/|-1|)^max(|-1/2|, 1/|-1/2|) + |-1/3|^max(|-1/2|, 1/|-1/2|)^max(|-1|, 1/|-1|)

= 1^max(1/2, 2)^max(1/3, 3) + 1^max(1/3, 3)^max(1/2, 2) + (1/2)^max(1, 1)^max(1/3, 3)

+ (1/2)^max(1/3, 3)^max(1, 1) + (1/3)^max(1, 1)^max(1/2, 2) + (1/3)^max(1/2, 2)^max(1, 1)

= 1^2^3 + 1^3^2 + (1/2)^1^3 + (1/2)^3^1 + (1/3)^1^2 + (1/3)^2^1

= 1^8 + 1^9 + (1/2)^1 + (1/2)^3 + (1/3)^1 + (1/3)^2

= 1⁸ + 1⁹ + 1/2¹ + 1/2³ + 1/3¹ + 1/3² = 1 + 1 + 1/2 + 1/8 + 1/3 + 1/9 = 221/72;

^||?^⁺²a = ^?^⁺²|a| = a ^||?^⁺ a = |a| ^?^⁺ |a| = a₁ ^||?^⁺ a₂ = |a₁| ^?^⁺ |a₂|

= |a₁|^max(|a₂|, 1/|a₂|) + |a₂|^max(|a₁|, 1/|a₁|)

= |a|^max(|a|, 1/|a|) + |a|^max(|a|, 1/|a|) = |a|^{max(|a|, 1/|a|)} + |a|^{max(|a|, 1/|a|)} = 2|a|^{max(|a|, 1/|a|)} ;

^||?^⁺³a = ^?^⁺³|a| = a ^||?^⁺ a ^||?^⁺ a = |a| ^?^⁺ |a| ^?^⁺ |a| = a₁ ^||?^⁺ a₂ ^||?^⁺ a₃ = |a₁| ^?^⁺ |a₂| ^?^⁺ |a₃|

= |a₁|^max(|a₂|, 1/|a₂|)^max(|a₃|, 1/|a₃|) + |a₁|^max(|a₃|, 1/|a₃|)^max(|a₂|, 1/|a₂|) + |a₂|^max(|a₁|, 1/|a₁|)^max(|a₃|, 1/|a₃|)

+ |a₂|^max(|a₃|, 1/|a₃|)^max(|a₁|, 1/|a₁|) + |a₃|^max(|a₁|, 1/|a₁|)^max(|a₂|, 1/|a₂|) + |a₃|^max(|a₂|, 1/|a₂|)^max(|a₁|, 1/|a₁|)

= |a|^max(|a|, 1/|a|)^max(|a|, 1/|a|) + |a|^max(|a|, 1/|a|)^max(|a|, 1/|a|) + |a|^max(|a|, 1/|a|)^max(|a|, 1/|a|)

+ |a|^max(|a|, 1/|a|)^max(|a|, 1/|a|) + |a|^max(|a|, 1/|a|)^max(|a|, 1/|a|) + |a|^max(|a|, 1/|a|)^max(|a|, 1/|a|)

= 6|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|) = 6|a|^ ²max(|a|, 1/|a|);

^||?^⁺²(-1/2) = ^?^⁺²|-1/2| = (-1/2) ^||?^⁺ (-1/2) = |-1/2| ^?^⁺ |-1/2| = (1/2) ^?^⁺ (1/2) = (-1/2)₁ ^||?^⁺ (-1/2)₂

= |(-1/2)₁|^max(|(-1/2)₂|, 1/|(-1/2)₂|) + |(-1/2)₂|^max(|(-1/2)₁|, 1/|(-1/2)₁|)

= |(-1/2)|^max(|(-1/2)|, 1/|(-1/2)|) + |(-1/2)|^max(|(-1/2)|, 1/|(-1/2)|)

= 2|(-1/2)|^max(|(-1/2)|, 1/|(-1/2)|)= 2(1/2)^max(1/2, 2) = 2(1/2)² = 1/2;

^||?^⁺³(-1/2) = ^?^⁺³|-1/2| = (-1/2) ^||?^⁺ (-1/2) ^||?^⁺ (-1/2) = |-1/2| ^?^⁺ |-1/2| ^?^⁺ |-1/2| = (1/2) ^?^⁺ (1/2) ^?^⁺ (1/2)

= (1/2)₁ ^?^⁺ (1/2)₂ ^?^⁺ (1/2)₃

= |(-1/2)₁|^max(|(-1/2)₂|, 1/|(-1/2)₂|)^max(|(-1/2)₃|, 1/|(-1/2)₃|) + |(-1/2)₁|^max(|(-1/2)₃|, 1/|(-1/2)₃|)^max(|(-1/2)₂|, 1/|(-1/2)₂|)

+ |(-1/2)₂|^max(|(-1/2)₁|, 1/|(-1/2)₁|)^max(|(-1/2)₃|, 1/|(-1/2)₃|) + |(-1/2)₂|^max(|(-1/2)₃|, 1/|(-1/2)₃|)^max(|(-1/2)₁|, 1/|(-1/2)₁|)

+ |(-1/2)₃|^max(|(-1/2)₁|, 1/|(-1/2)₁|)^max(|(-1/2)₂|, 1/|(-1/2)₂|) + |(-1/2)₃|^max(|(-1/2)₂|, 1/|(-1/2)₂|)^max(|(-1/2)₁|, 1/|(-1/2)₁|)

= |-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|) + |-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|)

+ |-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|) + |-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|)

+ |-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|) + |-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|)

= 6|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|) = 6(1/2)^max(1/2, 2)^max(1/2, 2)

= 6(1/2)^2^2 = 6(1/2)⁴ = 3/8.

Commutative Sign-Modulus-Power-Sum Maximum-Exponentiation

Idea

The principal idea of commutative sign-modulus-power-sum maximum-exponentiation is as follows:

for any set of real numbers, determine their moduli (absolute values);

if there are coinciding (equal) values among these moduli, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) values to artificially differentiate between them;

consider all the permutations of these already distinguishable values of all those moduli;

for every permutation of these values, compose their power tower using them in the order corresponding to this permutation and replacing each exponent smaller than 1 with its multiplicative inverse (reciprocal);

assign (attach) every power tower its base number sign, namely the sign of the initial (given) number whose modulus is used as the base of this power tower;

for all these permutations together, compose the sum of all these signed power towers;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative sign-modulus-power-sum maximum-exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative sign-modulus-power-sum maximum-exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative sign-modulus-power-sum maximum-exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative sign-modulus-power-sum maximum-exponential is denoted by

... °^||?^⁺ a °^||?^⁺ ... °^||?^⁺ b °^||?^⁺ ... °^||?^⁺ c °^||?^⁺ ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative sign-modulus-power-sum maximum-exponential is denoted by

Ε°^||?Σ_j∈J a_j = °^||?^⁺_j∈J a_j = Ε°^||?Σ{a_j | j ∈ J} = °^||?^⁺{a_j | j ∈ J}

= ... °^||?^⁺ a_j' °^||?^⁺ ... °^||?^⁺ a_j'' °^||?^⁺ ... °^||?^⁺ a_j''' °^||?^⁺ ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative sign-modulus-power-sum maximum-tetration is denoted by

°^||?^⁺ⁿa = a °^||?^⁺ a °^||?^⁺ ... °^||?^⁺ a

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a °^||?^⁺ b = (sign a)|a|^max(|b|, 1/|b|) + (sign b)|b|^max(|a|, 1/|a|) = (sign a)|a|^{max(|b|, 1/|b|)} + (sign b)|b|^{max(|a|, 1/|a|)} ;

a °^||?^⁺ b °^||?^⁺ c

= (sign a)|a|^max(|b|, 1/|b|)^max(|c|, 1/|c|) + (sign a)|a|^max(|c|, 1/|c|)^max(|b|, 1/|b|)

+ (sign b)|b|^max(|a|, 1/|a|)^max(|c|, 1/|c|) + (sign b)|b|^max(|c|, 1/|c|)^max(|a|, 1/|a|)

+ (sign c)|c|^max(|a|, 1/|a|)^max(|b|, 1/|b|) + (sign c)|c|^max(|b|, 1/|b|)^max(|a|, 1/|a|);

Ε°^||?Σ_{j∈{1, 2, 3}} a_j = Ε°^||?Σ{a_j | j ∈ {1, 2, 3}}

= °^||?^⁺{a_j | j ∈ {1, 2, 3}} = °^||?^⁺_{j∈{1, 2, 3}} a_j

= a₁ °^||?^⁺ a₂ °^||?^⁺ a₃

= (sign a₁)|a₁|^max(|a₂|, 1/|a₂|)^max(|a₃|, 1/|a₃|) + (sign a₁)|a₁|^max(|a₃|, 1/|a₃|)^max(|a₂|, 1/|a₂|)

+ (sign a₂)|a₂|^max(|a₁|, 1/|a₁|)^max(|a₃|, 1/|a₃|) + (sign a₂)|a₂|^max(|a₃|, 1/|a₃|)^max(|a₁|, 1/|a₁|)

+ (sign a₃)|a₃|^max(|a₁|, 1/|a₁|)^max(|a₂|, 1/|a₂|) + (sign a₃)|a₃|^max(|a₂|, 1/|a₂|)^max(|a₁|, 1/|a₁|);

(-2) °^||?^⁺ (-1/3)

= [sign(-2)]|-2|^max(|-1/3|, 1/|-1/3|) + [sign(-1/3)]|-1/3|^max(|-2|, 1/|-2|)

= [sign(-2)]|-2|^{max(|-1/3|, 1/|-1/3|)} + [sign(-1/3)]|-1/3|^{max(|-2|, 1/|-2|)}

= (-1)2^max(1/3, 1/(1/3)) + (-1)(1/3)^max(2, 1/2)

= - 2^{max(1/3, 3)} - (1/3)^{max(2, 1/2)} = - 2³ + - (1/3)² = - 73/9;

(-1) °^||?^⁺ (-1/2) °^||?^⁺ (-1/3)

= [sign(-1)]|-1|^max(|-1/2|, 1/|-1/2|)^max(|-1/3|, 1/|-1/3|) + [sign(-1)]|-1|^max(|-1/3|, 1/|-1/3|)^max(|-1/2|, 1/|-1/2|)

+ [sign(-1/2)]|-1/2|^max(|-1|, 1/|-1|)^max(|-1/3|, 1/|-1/3|) + [sign(-1/2)]|-1/2|^max(|-1/3|, 1/|-1/3|)^max(|-1|, 1/|-1|)

+ [sign(-1/3)]|-1/3|^max(|-1|, 1/|-1|)^max(|-1/2|, 1/|-1/2|) + [sign(-1/3)]|-1/3|^max(|-1/2|, 1/|-1/2|)^max(|-1|, 1/|-1|)

= (-1)1^max(1/2, 2)^max(1/3, 3) + (-1)1^max(1/3, 3)^max(1/2, 2)

+ (-1)(1/2)^max(1, 1)^max(1/3, 3) + (-1)(1/2)^max(1/3, 3)^max(1, 1)

+ (-1)(1/3)^max(1, 1)^max(1/2, 2) + (-1)(1/3)^max(1/2, 2)^max(1, 1)

= - 1^2^3 - 1^3^2 - (1/2)^1^3 - (1/2)^3^1 - (1/3)^1^2 - (1/3)^2^1

= - 1^8 - 1^9 - (1/2)^1 - (1/2)^3 - (1/3)^1 - (1/3)^2

= - 1⁸ - 1⁹ - 1/2¹ - 1/2³ - 1/3¹ - 1/3² = - 1 - 1 - 1/2 - 1/8 - 1/3 - 1/9 = -221/72;

°^||?^⁺²a = a °^||?^⁺ a = a₁ °^||?^⁺ a₂

= (sign a₁)|a₁|^max(|a₂|, 1/|a₂|) + (sign a₂)|a₂|^max(|a₁|, 1/|a₁|)

= (sign a)|a|^max(|a|, 1/|a|) + (sign a)|a|^max(|a|, 1/|a|)

= (sign a)|a|^{max(|a|, 1/|a|)} + (sign a)|a|^{max(|a|, 1/|a|)} = 2(sign a)|a|^{max(|a|, 1/|a|)} ;

°^||?^⁺³a = a °^||?^⁺ a °^||?^⁺ a = a₁ °^||?^⁺ a₂ °^||?^⁺ a₃

= (sign a₁)|a₁|^max(|a₂|, 1/|a₂|)^max(|a₃|, 1/|a₃|) + (sign a₁)|a₁|^max(|a₃|, 1/|a₃|)^max(|a₂|, 1/|a₂|)

+ (sign a₂)|a₂|^max(|a₁|, 1/|a₁|)^max(|a₃|, 1/|a₃|) + (sign a₂)|a₂|^max(|a₃|, 1/|a₃|)^max(|a₁|, 1/|a₁|)

+ (sign a₃)|a₃|^max(|a₁|, 1/|a₁|)^max(|a₂|, 1/|a₂|) + (sign a₃)|a₃|^max(|a₂|, 1/|a₂|)^max(|a₁|, 1/|a₁|)

= (sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|) + (sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|)

+ (sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|) + (sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|)

+ (sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|) + (sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|)

= 6(sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|) = 6(sign a)|a|^ ²max(|a|, 1/|a|);

°^||?^⁺²(-1/2) = (-1/2) °^||?^⁺ (-1/2) = (-1/2)₁ °^||?^⁺ (-1/2)₂

= [sign(-1/2)₁](-1/2)₁^max[|(-1/2)₂|, 1/|(-1/2)₂|] + [sign(-1/2)₂](-1/2)₂^max[|(-1/2)₁|, 1/|(-1/2)₁|]

= [sign(-1/2)](-1/2)^max[|(-1/2)|, 1/|(-1/2)|] + [sign(-1/2)](-1/2)^max[|(-1/2)|, 1/|(-1/2)|]

= (-1)(1/2)^max(1/2, 2) + (-1)(1/2)^max(1/2, 2) = - 2(1/2)^2 = - 2(1/2)² = -1/2;

°^||?^⁺³(-1/2) = (-1/2) °^||?^⁺ (-1/2) °^||?^⁺ (-1/2) = (-1/2)₁ °^||?^⁺ (-1/2)₂ °^||?^⁺ (-1/2)₃

= [sign(-1/2)₁]|(-1/2)₁|^max(|(-1/2)₂|, 1/|(-1/2)₂|)^max(|(-1/2)₃|, 1/|(-1/2)₃|)

+ [sign(-1/2)₁]|(-1/2)₁|^max(|(-1/2)₃|, 1/|(-1/2)₃|)^max(|(-1/2)₂|, 1/|(-1/2)₂|)

+ [sign(-1/2)₂]|(-1/2)₂|^max(|(-1/2)₁|, 1/|(-1/2)₁|)^max(|(-1/2)₃|, 1/|(-1/2)₃|)

+ [sign(-1/2)₂]|(-1/2)₂|^max(|(-1/2)₃|, 1/|(-1/2)₃|)^max(|(-1/2)₁|, 1/|(-1/2)₁|)

+ [sign(-1/2)₃]|(-1/2)₃|^max(|(-1/2)₁|, 1/|(-1/2)₁|)^max(|(-1/2)₂|, 1/|(-1/2)₂|)

+ [sign(-1/2)₃]|(-1/2)₃|^max(|(-1/2)₂|, 1/|(-1/2)₂|)^max(|(-1/2)₁|, 1/|(-1/2)₁|)

= [sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|) + [sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|)

+ [sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|) + [sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|)

+ [sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|) + [sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|)

= 6[sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|) = 6(-1)(1/2)^max(1/2, 2)^max(1/2, 2)

= - 6(1/2)^2^2 = - 6(1/2)⁴ = -3/8.

Commutative Sign-Power-Sum-Modulus Maximum-Exponentiation

Idea

The principal idea of commutative sign-power-sum-modulus maximum-exponentiation is as follows:

for any set of real numbers, determine their moduli (absolute values);

if there are coinciding (equal) values among these moduli, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) values to artificially differentiate between them;

consider all the permutations of these already distinguishable values of all those moduli;

for every permutation of these values, compose their power tower using them in the order corresponding to this permutation and replacing each exponent smaller than 1 with its multiplicative inverse (reciprocal);

assign (attach) every power tower its base number sign, namely the sign of the initial (given) number whose modulus is used as the base of this power tower;

for all these permutations together, compose the modulus (absolute value) of the sum of all these signed power towers;

in this sum, omit all those temporary attributes if they were artificially introduced.

Notata bene:

1. To provide the continuity of commutative sign-power-sum-modulus maximum-exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative sign-power-sum-modulus maximum-exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative sign-power-sum-modulus maximum-exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative sign-power-sum-modulus maximum-exponential is denoted by

... °^||?^^|+| a °^||?^^|+| ... °^||?^^|+| b °^||?^^|+| ... °^||?^^|+| c °^||?^^|+| ... = |... °^||?^⁺ a °^||?^⁺ ... °^||?^⁺ b °^||?^⁺ ... °^||?^⁺ c °^||?^⁺ ...|;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative sign-power-sum-modulus maximum-exponential is denoted by

Ε°^||?|Σ|_j∈J a_j = °^||?^^|+|_j∈J a_j = Ε°^||?|Σ|{a_j | j ∈ J} = °^||?^^|+|{a_j | j ∈ J}

= |Ε°^||?Σ_j∈J a_j| = |°^||?^⁺_j∈J a_j| = |Ε°^||?Σ{a_j | j ∈ J}| = |°^||?^⁺{a_j | j ∈ J}|

= ... °^||?^^|+| a_j' °^||?^^|+| ... °^||?^^|+| a_j'' °^||?^^|+| ... °^||?^^|+| a_j''' °^||?^^|+| ...

= |... °^||?^⁺ a_j' °^||?^⁺ ... °^||?^⁺ a_j'' °^||?^⁺ ... °^||?^⁺ a_j''' °^||?^⁺ ...|;

for any n ∈ N = {1, 2, 3, ...}, a commutative sign-power-sum-modulus maximum-tetration is denoted by

°^||?^^|+|na = |°^||?^⁺ⁿa| = a °^||?^^|+| a °^||?^^|+| ... °^||?^^|+| a = |a °^||?^⁺ a °^||?^⁺ ... °^||?^⁺ a|

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a °^||?^^|+| b = |a °^||?^⁺ b| = |(sign a)|a|^max(|b|, 1/|b|) + (sign b)|b|^max(|a|, 1/|a|)|

= |(sign a)|a|^{max(|b|, 1/|b|)} + (sign b)|b|^{max(|a|, 1/|a|)}|;

a °^||?^^|+| b °^||?^^|+| c = |a °^||?^⁺ b °^||?^⁺ c|

= |(sign a)|a|^max(|b|, 1/|b|)^max(|c|, 1/|c|) + (sign a)|a|^max(|c|, 1/|c|)^max(|b|, 1/|b|)

+ (sign b)|b|^max(|a|, 1/|a|)^max(|c|, 1/|c|) + (sign b)|b|^max(|c|, 1/|c|)^max(|a|, 1/|a|)

+ (sign c)|c|^max(|a|, 1/|a|)^max(|b|, 1/|b|) + (sign c)|c|^max(|b|, 1/|b|)^max(|a|, 1/|a|)|;

Ε°^||?|Σ|_{j∈{1, 2, 3}} a_j = Ε°^||?|Σ|{a_j | j ∈ {1, 2, 3}} = °^||?^^|+|{a_j | j ∈ {1, 2, 3}} = °^||?^^|+|_{j∈{1, 2, 3}} a_j

= |Ε°^||?Σ_{j∈{1, 2, 3}} a_j| = |Ε°^||?Σ{a_j | j ∈ {1, 2, 3}}| = |°^||?^⁺{a_j | j ∈ {1, 2, 3}}| = |°^||?^⁺_{j∈{1, 2, 3}} a_j|

= a₁ °^||?^^|+| a₂ °^||?^^|+| a₃ = |a₁ °^||?^⁺ a₂ °^||?^⁺ a₃|

= |(sign a₁)|a₁|^max(|a₂|, 1/|a₂|)^max(|a₃|, 1/|a₃|) + (sign a₁)|a₁|^max(|a₃|, 1/|a₃|)^max(|a₂|, 1/|a₂|)

+ (sign a₂)|a₂|^max(|a₁|, 1/|a₁|)^max(|a₃|, 1/|a₃|) + (sign a₂)|a₂|^max(|a₃|, 1/|a₃|)^max(|a₁|, 1/|a₁|)

+ (sign a₃)|a₃|^max(|a₁|, 1/|a₁|)^max(|a₂|, 1/|a₂|) + (sign a₃)|a₃|^max(|a₂|, 1/|a₂|)^max(|a₁|, 1/|a₁|)|;

(-2) °^||?^^|+| (-1/3) = |(-2) °^||?^^|+| (-1/3)|

= |[sign(-2)]|-2|^max(|-1/3|, 1/|-1/3|) + [sign(-1/3)]|-1/3|^max(|-2|, 1/|-2|)|

= |[sign(-2)]|-2|^{max(|-1/3|, 1/|-1/3|)} + [sign(-1/3)]|-1/3|^{max(|-2|, 1/|-2|)}|

= |(-1)2^max(1/3, 1/(1/3)) + (-1)(1/3)^max(2, 1/2)|

= |- 2^{max(1/3, 3)} - (1/3)^{max(2, 1/2)}| = |- 2³ + - (1/3)²| = |- 73/9| = 73/9;

(-1) °^||?^^|+| (-1/2) °^||?^^|+| (-1/3) = |(-1) °^||?^⁺ (-1/2) °^||?^⁺ (-1/3)|

= |[sign(-1)]|-1|^max(|-1/2|, 1/|-1/2|)^max(|-1/3|, 1/|-1/3|) + [sign(-1)]|-1|^max(|-1/3|, 1/|-1/3|)^max(|-1/2|, 1/|-1/2|)

+ [sign(-1/2)]|-1/2|^max(|-1|, 1/|-1|)^max(|-1/3|, 1/|-1/3|) + [sign(-1/2)]|-1/2|^max(|-1/3|, 1/|-1/3|)^max(|-1|, 1/|-1|)

+ [sign(-1/3)]|-1/3|^max(|-1|, 1/|-1|)^max(|-1/2|, 1/|-1/2|) + [sign(-1/3)]|-1/3|^max(|-1/2|, 1/|-1/2|)^max(|-1|, 1/|-1|)|

= |(-1)1^max(1/2, 2)^max(1/3, 3) + (-1)1^max(1/3, 3)^max(1/2, 2)

+ (-1)(1/2)^max(1, 1)^max(1/3, 3) + (-1)(1/2)^max(1/3, 3)^max(1, 1)

+ (-1)(1/3)^max(1, 1)^max(1/2, 2) + (-1)(1/3)^max(1/2, 2)^max(1, 1)|

= |- 1^2^3 - 1^3^2 - (1/2)^1^3 - (1/2)^3^1 - (1/3)^1^2 - (1/3)^2^1|

= |- 1^8 - 1^9 - (1/2)^1 - (1/2)^3 - (1/3)^1 - (1/3)^2|

= |- 1⁸ - 1⁹ - 1/2¹ - 1/2³ - 1/3¹ - 1/3²| = |- 1 - 1 - 1/2 - 1/8 - 1/3 - 1/9| = |-221/72| = 221/72;

°^||?^^|+|2a = a °^||?^^|+| a = a₁ °^||?^^|+| a₂

= |°^||?^⁺²a| = |a °^||?^⁺ a| = |a₁ °^||?^⁺ a₂|

= |(sign a₁)|a₁|^max(|a₂|, 1/|a₂|) + (sign a₂)|a₂|^max(|a₁|, 1/|a₁|)|

= |(sign a)|a|^max(|a|, 1/|a|) + (sign a)|a|^max(|a|, 1/|a|)|

= |(sign a)|a|^{max(|a|, 1/|a|)} + (sign a)|a|^{max(|a|, 1/|a|)}| = 2|(sign a)|a|^{max(|a|, 1/|a|)}| = 2|a|^{max(|a|, 1/|a|)};

°^||?^^|+|3a = a °^||?^^|+| a °^||?^^|+| a = a₁ °^||?^^|+| a₂ °^||?^^|+| a₃

= |°^||?^⁺³a| = |a °^||?^⁺ a °^||?^⁺ a| = |a₁ °^||?^⁺ a₂ °^||?^⁺ a₃|

= |(sign a₁)|a₁|^max(|a₂|, 1/|a₂|)^max(|a₃|, 1/|a₃|) + (sign a₁)|a₁|^max(|a₃|, 1/|a₃|)^max(|a₂|, 1/|a₂|)

+ (sign a₂)|a₂|^max(|a₁|, 1/|a₁|)^max(|a₃|, 1/|a₃|) + (sign a₂)|a₂|^max(|a₃|, 1/|a₃|)^max(|a₁|, 1/|a₁|)

+ (sign a₃)|a₃|^max(|a₁|, 1/|a₁|)^max(|a₂|, 1/|a₂|) + (sign a₃)|a₃|^max(|a₂|, 1/|a₂|)^max(|a₁|, 1/|a₁|)|

= |(sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|) + (sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|)

+ (sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|) + (sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|)

+ (sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|) + (sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|)|

= |6(sign a)|a|^max(|a|, 1/|a|)^max(|a|, 1/|a|)| = 6|a|^ ²max(|a|, 1/|a|);

°^||?^^|+|2(-1/2) = (-1/2) °^||?^^|+| (-1/2) = (-1/2)₁ °^||?^^|+| (-1/2)₂

|°^||?^⁺²(-1/2)| = |(-1/2) °^||?^⁺ (-1/2)| = |(-1/2)₁ °^||?^⁺ (-1/2)₂|

= |[sign(-1/2)₁](-1/2)₁^max[|(-1/2)₂|, 1/|(-1/2)₂|] + [sign(-1/2)₂](-1/2)₂^max[|(-1/2)₁|, 1/|(-1/2)₁|]|

= |[sign(-1/2)](-1/2)^max[|(-1/2)|, 1/|(-1/2)|] + [sign(-1/2)](-1/2)^max[|(-1/2)|, 1/|(-1/2)|]|

= |(-1)(1/2)^max(1/2, 2) + (-1)(1/2)^max(1/2, 2)| = |- 2(1/2)^2| = |- 2(1/2)²| = |-1/2| = 1/2;

^±||?^^|+|3(-1/2) = (-1/2) ^±||?^^|+| (-1/2) ^±||?^^|+| (-1/2) = (-1/2)₁ ^±||?^^|+| (-1/2)₂ ^±||?^^|+| (-1/2)₃

|°^||?^⁺³(-1/2)| = |(-1/2) °^||?^⁺ (-1/2) °^||?^⁺ (-1/2)| = |(-1/2)₁ °^||?^⁺ (-1/2)₂ °^||?^⁺ (-1/2)₃|

= |[sign(-1/2)₁]|(-1/2)₁|^max(|(-1/2)₂|, 1/|(-1/2)₂|)^max(|(-1/2)₃|, 1/|(-1/2)₃|)

+ [sign(-1/2)₁]|(-1/2)₁|^max(|(-1/2)₃|, 1/|(-1/2)₃|)^max(|(-1/2)₂|, 1/|(-1/2)₂|)

+ [sign(-1/2)₂]|(-1/2)₂|^max(|(-1/2)₁|, 1/|(-1/2)₁|)^max(|(-1/2)₃|, 1/|(-1/2)₃|)

+ [sign(-1/2)₂]|(-1/2)₂|^max(|(-1/2)₃|, 1/|(-1/2)₃|)^max(|(-1/2)₁|, 1/|(-1/2)₁|)

+ [sign(-1/2)₃]|(-1/2)₃|^max(|(-1/2)₁|, 1/|(-1/2)₁|)^max(|(-1/2)₂|, 1/|(-1/2)₂|)

+ [sign(-1/2)₃]|(-1/2)₃|^max(|(-1/2)₂|, 1/|(-1/2)₂|)^max(|(-1/2)₁|, 1/|(-1/2)₁|)|

= |[sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|) + [sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|)

+ [sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|) + [sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|)

+ [sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|) + [sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|)|

= |6[sign(-1/2)]|-1/2|^max(|-1/2|, 1/|-1/2|)^max(|-1/2|, 1/|-1/2|)| = |6(-1)(1/2)^max(1/2, 2)^max(1/2, 2)|

= |- 6(1/2)^2^2| = |- 6(1/2)⁴| = |-3/8| = 3/8.

Commutative Modulus-Power-Sum-Sign Maximum-Exponentiation

Idea

The principal idea of commutative modulus-power-sum-sign maximum-exponentiation is as follows:

for any set of real numbers, determine their moduli (absolute values);

if there are coinciding (equal) values among these moduli, then temporarily assign (attach) any suitable distinguishing attributes, e.g. indices, to these coinciding (equal) values to artificially differentiate between them;

consider all the permutations of these already distinguishable values of all those moduli;

for every permutation of these values, compose their power tower using them in the order corresponding to this permutation and replacing each exponent smaller than 1 with its multiplicative inverse (reciprocal);

for all these permutations together, compose the sum of all these power towers;

in this sum, omit all those temporary attributes if they were artificially introduced;

assign (attach) this sum the sign of the sum of all the initial (given) numbers.

Notata bene:

1. To provide the continuity of commutative modulus-power-sum-sign maximum-exponential functions as all or some of their arguments approach their common (equal) values, it is necessary and sufficient to completely regard all (also equal) power towers whose total number has to equal the total number of the corresponding permutations as if they were without repetitions. Namely, the total number of different permutations of n distinct objects is n! (n factorial).

2. To avoid possible mistakes whenever the values of all or some arguments of commutative modulus-power-sum-sign maximum-exponential functions coincide, it can be useful to temporarily assign (attach) any suitable artificial distinguishing attributes, e.g. indices, to such equal values to artificially differentiate between them, to further compose the corresponding sums, and to then omit these artificial attributes.

Notation

Let us denote the commutative modulus-power-sum-sign maximum-exponential of real numbers as follows:

for any set of real numbers ... , a , ... , b , ... , c , ... , their commutative modulus-power-sum-sign maximum-exponential is denoted by

... ^||?^⁺° a ^||?^⁺° ... ^||?^⁺° b ^||?^⁺° ... ^||?^⁺° c ^||?^⁺° ... ;

for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their commutative modulus-power-sum-sign maximum-exponential is denoted by

Ε^||?°Σ_j∈J a_j = ^||?^^+±_j∈J a_j = Ε^||?±Σ{a_j | j ∈ J} = ^||?^⁺°{a_j | j ∈ J}

= ... ^||?^⁺° a_j' ^||?^⁺° ... ^||?^⁺° a_j'' ^||?^⁺° ... ^||?^⁺° a_j''' ^||?^⁺° ... ;

for any n ∈ N = {1, 2, 3, ...}, a commutative modulus-power-sum-sign maximum-tetration is denoted by

^||?^⁺°ⁿa = a ^||?^⁺° a ^||?^⁺° ... ^||?^⁺° a = sign(na)[|a| ^?^⁺ |a| ^?^⁺ ... ^?^⁺ |a|] = sign(na)^?^⁺ⁿ|a|

with a used n times on the right-hand side.

Sense (Meaning) and Examples

a ^||?^⁺° b = sign(a + b)(a ^||?^⁺ b) = sign(a + b)(|a| ^?^⁺ |b|)

= sign(a + b)[|a|^max(|b|, 1/|b|) + |b|^max(|a|, 1/|a|)] = sign(a + b)[|a|^{max(|b|, 1/|b|)} + |b|^{max(|a|, 1/|a|)}];

a ^||?^⁺° b ^||?^⁺° c = sign(a + b)(|a| ^?^⁺ |b| ^?^⁺ |c|)

= sign(a + b)[|a|^max(|b|, 1/|b|)^max(|c|, 1/|c|) + |a|^max(|c|, 1/|c|)^max(|b|, 1/|b|) + |b|^max(|a|, 1/|a|)^max(|c|, 1/|c|)

+ |b|^max(|c|, 1/|c|)^max(|a|, 1/|a|) + |c|^max(|a|, 1/|a|)^max(|b|, 1/|b|) + |c|^max(|b|, 1/|b|)^max(|a|, 1/|a|)];

Ε^||?°Σ_{j∈{1, 2, 3}} a_j = Ε^||?°Σ{a_j | j ∈ {1, 2, 3}} = ^||?^⁺°_{j∈{1, 2, 3}} a_j = ^||?^⁺°{a_j | j ∈ {1, 2, 3}}

= sign(Σ_{j∈{1, 2, 3}} a_j)Ε^?Σ_{j∈{1, 2, 3}} |a_j| = sign({Σ|a_j| | j ∈ {1, 2, 3}})Ε^?Σ{|a_j| | j ∈ {1, 2, 3}}

= sign(a₁ + a₂ + a₃)^?^⁺_{j∈{1, 2, 3}} |a_j| = sign(a₁ + a₂ + a₃)^?^⁺{|a_j| | j ∈ {1, 2, 3}}

= a₁ ^||?^⁺° a₂ ^||?^⁺° a₃ = sign(a₁ + a₂ + a₃)[|a₁| ^?^⁺ |a₂| ^?^⁺ |a₃|]

= sign(a₁ + a₂ + a₃)[|a₁|^max(|a₂|, 1/|a₂|)^max(|a₃|, 1/|a₃|) + |a₁|^max(|a₃|, 1/|a₃|)^max(|a₂|, 1/|a₂|) + |a₂|^max(|a₁|, 1/|a₁|)^max(|a₃|, 1/|a₃|)

+ |a₂|^max(|a₃|, 1/|a₃|)^max(|a₁|, 1/|a₁|) + |a₃|^max(|a₁|, 1/|a₁|)^max(|a₂|, 1/|a₂|) + |a₃|^max(|a₂|, 1/|a₂|)^max(|a₁|, 1/|a₁|)];