2000 MSC primary 00A05; sec. 00A69, 00A71, 03E99
Quantiintervals and Semiquantiintervals (Essential)
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
RUAG Aerospace Services GmbH, Germany
Mathematical Journal
of the "Collegium" All World Academy of Sciences
Munich (Germany)
5 (2005), 2
Quantiintervals and directed ones of any (e.g. infinite and/or negative) length with any quantities of their bounds are particular quantisets. To briefly write many results of the same type, we shall introduce the initial and final semiquantiintervals (combining them) and a general notation of their brackets.
Definition 1. An initial and a final semiquantiintervals are symbol combinations of the forms )qa T and T rb(, respectively, where the bounds a and b and their quantities q and r belong to a uninumber set S, the brackets ) and ( are any [possibly the same] elements of the set {] , | , [}, and the extension T is an ordered subset of S.
Definition 2. A quantiinterval as the commutative quantiunion of an initial semiquantiinterval and a final one, both having a common extension, say )qa T rb) =° )qa T ∪° T rb) =° T rb) ∪° )qa T , is a quantiset containing both the ordinary set ]a T b[ ⊆ T of all intermediate uninumbers t ∈ T , for which either a < t < b or a > t > b, and each bound with its own quantity multiplied by 0 [always for a , b ∉ T], 1/2, or 1 in accordance with the adjacent bracket [with it, a bound has to be preserved even if its quantity vanishes: (01 T ≠° (02 T ≠° (0# T ]:
]qa T =° ]a T ; if a ∈ T , |qa T =° q/2a∪° ]a T , [qa T =° qa∪° ]a T ;
T rb[ =° T b[ ; if b ∈ T , T rb| =° T b[ ∪° r/2b , T rb] =° T b[ ∪° rb .
Corollary 3. The rearrangement of the bounds with their quantities by preserving the relative orientation of each bracket with reference to the adjacent bound of a quantiinterval does not affect it:
)qa T rb) =° (rb T qa( , say [qa T rb| =° |rb T qa].
Definition 4. Open, some partially open (partially closed), in particular, half-open (half-closed), and closed quantiintervals of zero length are for a ∈ T (for a ∉ T they all are empty 0# =° ∅):
]a T a[ =°-1a ; |a T a[ =° ]a T a| =° -1/2a ; |a T a] =° [a T a| =° 1/2a ;
[a T a[ =° |a T a| =° ]a T a] =° 0a =° 0# =° ∅ ; [a T a] =° 1a =° {a}.
Definition 5. For any a , b , q , r ∈ R°, T ∈ {R , R°}:
(1) a real quantiinterval is )qa , rb( =° )qa R rb( , say a symmetric half-open real quantiinterval |a , b| =° 1/2a ∪° ]a , b[ ∪° 1/2b ;
(2) a quantireal quantiinterval is )qa ,° rb( =° )qa R° rb( , say
)a ,° b(=° ∪°d ∈ )a , b(d [d the monad of d];
(3) the following quantireal semiquantiintervals with a ≤ b to be combined are [a+, a- positive and negative submonads of a]:
[a,° =° {a} ∪° a+ ∪° ]a ,° ; |a ,° =° |a ,° =° 1/2a ∪° a+ ∪° ]a ,° ;
]a ,° =° a+ ∪° ]a ,° ; , °b] =° , °b[∪° b- ∪° {b} ;
, ° b| =° , °b| =° , °b[ ∪° b- ∪° 1/2b ; ,° b[ =° , °b[ ∪° b- .
Definition 6. A directed quantiinterval, say )qa T rb) [the italic brackets, a the origin, b the end, q and r the quantities of these bounds], is a quantiset that consists of the elements of its generating quantiinterval )qa T rb) [the relative orientation of each bracket with reference to the adjacent bound is preserved] with the same (opposite) quantities if a ≤ b (a > b , respectively):
)qa T rb) =° )qa T rb) if a ≤ b, )qa T rb) =° -1)qa T rb) if a > b .
Corollary 7. The rearrangement of the distinct bounds with their own quantities by preserving the relative orientation of each bracket with reference to the adjacent bound of a directed quantiinterval implies its additive inversion:
)qa T rb) =° -1(rb T qa( , say [qa T rb| =° -1|rb T qa].
Notation 8. The complete algebraic additivity of a quantiset correspondence formally applies to the semiquantiintervals, quantiintervals, and directed quantiintervals, say:
f (|a,) = f(1/2a) + f(]a ,) = 1/2f(a) + f(]a ,).
The introduced
quantiintervals
and directed ones both based on semiquantiintervals apply to information and other problems.