## Idea ## A ternary function which behaves as simultaneous actions on both the left and the right side: ## Definition ## Given a set $S$ and monoids $(M, e_M, \mu_M)$ and $(N, e_N, \mu_N)$, a two-sided action is a ternary function $\alpha:M \times S \times N \to S$ such that * for all $s:S$, $\alpha(e_M, s, e_N) = s$ * for all $s:S$, $a:M$, $b:M$, $c:N$, and $d:N$, $\alpha(a, \alpha(b, s, c), d) = \alpha(\mu_M(a, b), s, \mu_N(c, d))$ The left action is defined as $$\alpha_L(a, s) \coloneqq \alpha(a, s, e_N)$$ for $a:M$ and $s:S$, and the right action is defined as $$\alpha_R(s, c) \coloneqq \alpha(e_M, s, c)$$ for $c:N$ and $s:S$. ## See also ## * [[Malcev operation]] * [[action]] * [[bimodule]]