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 = ...