#Contents# * table of contents {:toc} ## Definition ## Let $(M, e, \mu)$ be a [[monoid]] and let $X$ be a [[set]]. An __action__ is a function $\alpha_f: M \to (X \to X)$ where there exist an identity term $$i_f: \alpha_f(e) = id_X$$ and dependent function $$c_f: \prod_{g:M} \prod_{h:M} \alpha_f(g) \circ \alpha_f(h) = \alpha_f(\mu(g, h))$$ ## See also ## * [[monoid]] * [[module]]