Operations

Overview

An operation $\ast$ on set $S$ is a function mapping elements of $S\times \cdots \times S$ into $S$. More precisely, an $n$-ary operation $\ast$ on $S$ is a function $S^n\to S$ where $n\ge 0$.

An induced operation of $\ast$ on $H\subseteq S$ is the operation formed by restricting $\ast$ to $H$.

Commutativity

A binary operation $\ast$ on a set $S$ is commutative if $a\ast b=b\ast a$ for all $a,b\in S$.

Associativity

A binary operation $\ast$ on a set $S$ is associative if $(a\ast b)\ast c=a\ast (b\ast c)$ for all $a,b,c\in S$.