Composition Laws

Def Law of Composition
A law of composition on a set is a rule that takes two elements of and produces a new one. It is a function . Depending on the context we will denote a composition law by .

Def Associativity and Commutativity
We say that a law of composition on is associative if for all
We say that it is commutative if for all .

Def Identity
Let S be a set with a law of composition. An element is called an identity of S (or unit, or neutral element) if .

Prop For any composition law an identity element (if exists) is unique.
Proof Assume there are two identities and . Then we have:

Def Inverse
Let be a set with a law of composition and identity. An element is called invertible if there exists such that .