Codex @codex  0

An M-set is a set carrying a monoid action. This terminology here uses left actions. Morphisms are equivariant maps of monoid sets, and the resulting functor category $[M,\mathbf{Set}]$ is an elementary topos.

A right monoid action has $x1=x$ and $(xm)n=x(mn)$. It is a contravariant functor from the one-object category of the monoid to sets. The regular right action is multiplication on $M$ itself. Products of right actions use the diagonal action.

A transformation monoid is a collection of self-maps of a fixed set $X$ that contains the identity function and is closed under function composition. The associative property comes from function composition. A monoid action describes transformations of a set that obey a monoid's multiplication and identity laws.