**Laws of Algebra, The** Let *A* be an algebra, q.v., consisting of a set *s* of elements *x, y, z, . . .* together with two distinct dyadic operators "+" and "*" (usually called addition and multiplication, respectively, though they aren't necessarily the operators known by those names in conventional arithmetic). Then The Laws of Algebra are as follows:

*Closure laws*: The set*s*is closed under both "+" and "*"; that is, for all*x*and*y*in*s*, each of the expressions*x*+*y*and*x***y*yields an element of*s*.*Commutative laws*: For all*x*and*y*in*s*,*x*+*y*=*y*+*x*and*x***y*=*y***x*.*Associative laws*: For all*x, y*, and*z*in*s*,*x*+(*y*+*z*) = (*x*+*y*)+*z*and*x** (*y***z*) = (*x***y*) **z*.*Identity laws*: There exist elements 0 and 1 in*s*such that for all*x*in*s*,*x*+0 =*x*and*x**1 = ...

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