What do we call functions satisfying $[a[b]c] = [abc]$? Let $M$ denote a monoid and suppose we're given a function $[-] : M \rightarrow M$ satisfying $[a[b]c] = [abc].$ Then:

Proposition 0. $[-]$ is idempotent.

Proof. Take $a=c=1$).

Proposition 1. The set of fixed points of $[-]$ becomes a monoid with identity $[1]$ and multiplication $a,b \mapsto [ab]$.

Proof.
Associativity: $[[ab]c] = [abc] = [a[bc]]$
Left-identity: $[[1]a] = [1a] = [a] = a$
Right-identity: $[a[1]] = [a1] = [a] = a$

Question. What are functions satisfying this condition called?

I'm also interested in higher-categorical generalizations.
 A: Here is a simple observation: The condition is equivalent to $$\forall a,b \in M. \, [a \cdot [b]]=[a \cdot  b]=[[a] \cdot  b].$$
Assume that $M$ is a preordered monoid. Then it is natural to assume $a \leq [a]$, and $a \mapsto [a]$ behaves like a "closure operator". The fixed points are the closed elements. There are lots of examples for this. This setting can be generalized and has been studied before:
Assume that $M$ is a monoidal category with underlying category $C$ and $R : C \to C$ is a functor equipped with a natural transformation $\eta : \mathrm{id}_C \to R$ (satisfying $R\eta=\eta R$). Then one may demand that the induced morphisms
$$R(a \otimes R(b)) \leftarrow R(a \otimes b)  \to R(R(a) \otimes b)$$
are isomorphisms. This situation appears in Day's reflection theorem for closed monoidal categories; here the reflection is called normal. This is used to endow reflective subcategories of $C$ with a monoidal structure. It is also useful for the construction of monoidal localizations, see Day's Note on monoidal localization. I would also consult the papers which cite these.
A: Here is another notion. Let ' be the operation of adding a unit to a monoid.  Let [] be its inverse, which is the identity, except the old and new unit map to the old unit.  This is another example of how things behave. This suggests to me that [] behaves well on congruences of the monoid, and that there may be some polynomial operations that can represent [].
Gerhard "Make This More Generally Algebraic" Paseman, 2016.12.29.
