2000 MSC prim. 00A71; sec. 03E10, 12D99, 26E30, 28A75
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)
4(2004), 1
Fractional quantisets are introduced to extend modeling.
Theorem 1. If the quantities q in some quantielements qa with the same basis a build a (possibly commutative) additive group with zero 0 and the inverse -q to q , then the quantielements build a (commutative, respectively) additive group with zero 0# (# empty element) and inverse -qa and the two groups are isomorphic with the q -- qa isomorphism.
Theorem 2. If both the bases a and the quantities q in some quantielements build two (commutative) multiplicative groups with the units u and 1 and inverses u/a and 1/q, then the quantielements build a (commutative, respectively) multiplicative group with the unit 1u and the inverse 1/q(u/a) and the correspondence (a, q) -- qa between these three groups is bijective und homomorphic.
Corollary 3. If the distributive law for the quantities q holds, the same holds for the quantielements with the same basis a.
Theorem 4. If the quantities s in some quantielements with the unit basis u build a (possibly commutative) field, then the same holds for the quantielements.
Corollary 5. If in Theorem 4 the quantities in some quantielements qa with the same basis a build a commutative ring containing the field of quantities s, then scalars su and vectors qa build a vector space.
Definition 6. A general union and intersection mutually distributive of some quantielements with the same basis and ordered quantities are
... + qa + ... + qa + ... = sup{... ,q , ... , r , ...}a ,
... * qa * ... * qa * ... = inf{... ,q , ... , r , ...}a .
Corollary 7. Quantielements qa with the same basis build a distributive algebra (not always complementary/Boolean). If the set of q has its greatest lower inf{q} and/or least upper bound sup{q}, then this algebra has zero inf{q}a and/or the unit sup{q}a. If and only if the set has the both bounds, then this zero and this unit only are mutually complementary.
Definition 8. A (distributive) algebra is called extremely complementary (extremely Boolean) if it has zero, unit, and a complement to each element extreme by an order.
Corollary 9. If the bases a and quantities q in some quantisets build a (commutative) multiplicative semigroup S and a commutative ring R, then the quantisets build a (commutative) ring R. If S and R have units, R is a ring with unit and can have zero divisors even if R is free of those.
Corollary 10. If in Corollary 9 the quantities in some quantielements-scalars su build a (commutative) ring K including in R and R is commutative, then these scalars and the quantisets-vectors build a (commutative) algebra.
Corollary 11. Reduced quantisets A with ordered quantities of each basis build (with the general union and intersection) a distributive algebra (not always complementary and Boolean). If the set of the quantities for each basis a has its greatest lower inf({q}, a) and/or least upper bound sup({q}, a), then this algebra has zero inf A =° {... , inf({q}, a)a, ...}° and/or unit sup A =° {... , sup({q}, a)a, ...}°. If and only if A has the both bounds, these quantielements only are mutually complementary and an extreme quantiset (with extreme quantities for each basis) only has a unique complement.
Corollary 12. The integral (fractional) quantisets form with general addition and multiplication a ring with unit and an algebra (a field and an algebra, respectively), with general unification and intersection a so-called extremely Boolean algebra (a distributive algebra, respectively) very useful for solving urgent scientific and life problems, e.g. coding ones.