3
$\begingroup$

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.

Improve this question
$\endgroup$
2
  • $\begingroup$ Do you have examples in mind? E.g. [f]=f(0), [M]=det(M), [z]=|z|, [(x1,...xn)]=([x1],...[xn])? $\endgroup$
    – user44143
    Commented Dec 29, 2016 at 17:24
  • $\begingroup$ Another example is the additive monoid of ideals of a commutative ring and the operation of taking the radical: We have $\displaystyle\sqrt{I+J} = \sqrt{I+\sqrt{J}}$. Prop 1 tells us that there is an additive monoid of radical ideals, where the "radical sum" of two radical ideals $I,J$ is $\sqrt{I+J}$. $\endgroup$
    – HeinrichD
    Commented Dec 30, 2016 at 0:57

2 Answers 2

Reset to default
6
$\begingroup$

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.

Improve this answer
$\endgroup$
0
$\begingroup$

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.

Improve this answer
$\endgroup$
2
  • 1
    $\begingroup$ Not quite sure I follow. Can you elaborate a little? $\endgroup$
    – goblin GONE
    Commented Dec 30, 2016 at 3:01
  • $\begingroup$ The [] operation reminded me of some similar functions used in the beginnings of tame congruence theory (TCT), and a simple version of this in a monoid context is "undoing" an adjoined unit. One can imagine an iterated version undoing a succession of adjoined units, and this suggested to me operations on congruence classes, which also appears in TCT. Rather than suggest "Find an answer in TCT" (I think you won't), I suggest "Look among congruence preserving polynomial operations", where you might find a name for your relation. Gerhard "Can't Elaborate Too Much More" Paseman, 2016.12.29. $\endgroup$
    – Gerhard Paseman
    Commented Dec 30, 2016 at 3:25

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .