# Action

In abstract algebra an action of a set $\Omega$ on a set $S$ is a mapping $\alpha \mapsto f_\alpha$ of $\Omega$ into $S^S$, the set of functions of $S$ into itself. When there is no risk of confusion, the element $f_\alpha (x)$, for $x\in S$ and $\alpha \in \Omega$, is sometimes denoted $\alpha x$, or $x \alpha$.

Let $\Omega, S,T$ be sets, and let $\alpha \mapsto f_\alpha$ and $\alpha \mapsto g_\alpha$ be actions of $\Omega$ on $S$ and $T$, respectively. An $\Omega$-morphism of $S$ into $T$ is a function $h: S \to T$ for which $h \circ f_\alpha = g_\alpha \circ h$, for all $\alpha$ in $\Omega$.

Let $\Omega, \Xi, S,T$ be sets, $\phi$ a function of $\Omega$ into $\Xi$, $\alpha \mapsto f_\alpha$ an action of $\Omega$ on $S$, and $\alpha \mapsto g_\alpha$ an action of $\Xi$ on $T$. A mapping $h: S \to T$ is called a $\phi$-morphism if $$(h \circ f_\alpha)(x) = (g_{\phi(\alpha)} \circ h)(x) ,$$ for all $\alpha$ in $\Omega$ and $x$ in $E$. If $\phi$ is the identity map of $\Omega$, then the terms "$\phi$-morphism" and "$\Omega$-morphism" are synonymous.