I came up with the following...
...Definition.
Let $(S,\wedge,1_S)$ denote a fixed but arbitrary monoid.
(In the examples I have in mind, $S$ is always commutative and idempotent. But the definition makes sense irrespective.)
Then by an "$S$-monoid," let us mean a monoid $(M,\circ,1_M)$ together with:
- a homomorphism $S \rightarrow M,$ denoted $s \mapsto \overline{s}$
- an action $M \times S \rightarrow S,$ denoted $m,s \mapsto m \star s,$ satisfying the identities $$m \star (s \wedge s') = (m\star s) \wedge (m\star s') \;\;\mbox{ and }\;\; m\star 1_S = 1_S$$
satisfying a "deformed" (and weakened) version of commutativity:
$$m \circ \overline{s} = \overline{m \star s} \circ m$$
Main Example. Let $(C,\wedge,1_C)$ denote a commutative monoid. Write $\mathrm{Idem}(C)$ for the subset of idempotent elements of $C$. Then $\mathrm{Idem}(C)$ is itself a monoid (and necessarily commutative and idempotent; not that it matters here.)
I claim $\mathrm{End}(C)$ can be viewed as an $\mathrm{Idem}(C)$-monoid as follows:
Given $s \in \mathrm{Idem}(C)$, we define $\overline{s} : C \rightarrow C$ by left multiplication:
$$\overline{s}(c) = s \wedge c$$
Given $m \in \mathrm{End}(C)$ and $s \in \mathrm{Idem}(C)$, we define $m \star s$ by evaluation:
$$m\star s = m(s)$$
(This works because endomorphisms preserve the property of being idempotent, so $m(s)$ is always idempotent, given that $s$ is.)
The verification of deformed commutativity is straightforward. Consider $m \in \mathrm{End}(C)$, $s \in \mathrm{Idem}(C)$ and $c \in C$. Then: $$(m \circ \overline{s})(c) = m(\overline{s}(c)) = m(s \wedge c) = m(s) \wedge m(c)$$
$$= (m \star s) \wedge m(c) = \overline{m \star s}(m(c)) = (\overline{m \star s} \circ m)(c)$$
We conclude that $$m \circ \overline{s} = \overline{m \star s} \circ m,$$ as required.
Question. Are $S$-monoids known to be good for anything? If so, what are they called in the literature, and where can I learn more about them?