# Q u a n t I s e t s a L g e b r a

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СМКЭС-2004

2000 MSC prim. 00A71; sec. 03E10, 12D99, 26E30, 28A75

## Q U A N T I S E T S A L G E B R A

L.G.Gelimson, Ph.D., D.Sc., RUAG Munich

iasco@web.de

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 an (a commutative, respectively) additive group with zero 0# (# the empty element) and the inverse -qa, and the two groups are isomorphic with the natural isomorphism q -- qa.

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, as well as the 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 natural correspondence (a, q) -- qa between these three groups is bijective und homomorphic.

Corollary 3. If the distributive law holds for the quantities q, 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 the quantities s, then the scalars su and the 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 +...+ra +...=sup{...,q, ..., r, ...}a, ...*qa *...*ra *...=inf{..., q,..., r, ...}a.

Corollary 7. Quantielements qa with the same basis build a distributive algebra (not always complementary or Boolean). If the set of all quantities q has its greatest lower inf{q} and/or least upper bound sup{q}, 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. An (A distributive) algebra is called extremely complementary (extremely Boolean, respectively) if it has zero, unit, and complements to all elements extreme by 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, respectively) ring R. If S and R have units, then 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, respectively) algebra.

Corollary 11. Reduced quantisets A with ordered quantities of each basis build (with 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 quantiset A has the both bounds, then these quantielements only are mutually complementary and an extreme quantiset (with the extreme quantities only 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.

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