Composition Laws

Def Law of Composition A law of composition on a set $S$ is a rule that takes two elements of $S$ and produces a new one. It is a function $S\times S\to S$. Depending on the context we will denote a composition law by $(a,b)\mapsto a\circ b,a\ast b,a+b,ab$.

Def Associativity and Commutativity We say that a law of composition on $S$ is associative if $(ab)c=a(bc)$ for all $a,b,c\in S$ We say that it is commutative if $ab=ba$ for all $a,b\in S$.

Def Identity Let S be a set with a law of composition. An element $e\in S$ is called an identity of S (or unit, or neutral element) if $ae=ea=a,\forall a\in S$.

Prop For any composition law an identity element (if exists) is unique. Proof Assume there are two identities $e_1$ and $e_2$. Then we have:$e_1=e_1{e}_2=e_2$

Def Inverse Let $S$ be a set with a law of composition and identity. An element $a\in S$ is called invertible if there exists $b\in S$ such that $ab=ba=e$.