Action

From AoPSWiki

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.