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Abstract

Laws of Algebra, The Let A be an algebra, q.v., consisting of 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 expression x+y and x*y yields an elements 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=x. The elements 0 and 1 are called the additive identity and the multiplicative identity, respectively.

• Inverse laws: For all x in s, there exist elements −x and (unless x=0 1/x in s such that x+(−x)=0 and x*(1/x)=1. The elements −x and 1/x are called the additive inverse and the multiplicative inverse, respectively (of x in each case). The expressions x+(−y) and x*(1/y) are usually abbreviated to x-y and x/y, respectively.

• Distributive law (of “*” over “+”): For all x, y, and z in s, x *(y+z)=(x*y)+(x*z).