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



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> **Proposition 0.** $[-]$ is idempotent.
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*Proof.* Take $a=c=1$).



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


Associativity: $[[ab]c] = [abc] = [a[bc]]$


Left-identity: $[[1]a] = [1a] = [a] = a$


Right-identity: $[a[1]] = [a1] = [a] = a$



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> **Question.** What are functions satisfying this condition called?
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I'm also interested in higher-categorical generalizations.



What do we call functions satisfying $[a[b]c] = [abc]$?
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](http://www.sciencedirect.com/science/article/pii/0022404972900217) 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](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0004972700045433). I would also consult the papers which cite these.


