# Commutativity

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## Commutativity

a property of the addition and multiplication of numbers expressed by the identities a + b = b + a and ab = ba. In a more general sense, the operation a * b is termed commutative if a * b = b *a. Addition and multiplication of polynomials, for example, have the property of commutativity; vector multiplication (see VECTOR PRODUCT) is not commutative since [a,b] = — [b,a].

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We have a minimal generating set for [A.sup.n-2.sub.1] [([T.sub.n], [T.sub.0]).sup.ab] and we can see that there are no relations between the generators outside of commutivity. Theorem 1.1 follows; that is, [A.sup.n-2.sub.1] [([T.sub.n], [T.sub.0]).sup.ab] is free abelian and of rank ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).
For instance, defining properties of commutivity and transitivity for a user object provides the compiler with more information to perform useful optimizations.
P(Cold) = [[Sigma].sub.Scratches,Cat,Sneeze,Allergy] P(Scratches \ Cat) * P(Sneeze \ Cold, Allergy) * P(Allergy \ Cat) * P(Cold) * P(Cat) The sum and product operations being performed here are real-number operations, so the normal associativity, commutivity, and distributivity properties apply.

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