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Let $A$ and $B$ be monoids, let $f\colon A\to B$ be a morphism of monoids. The following pair of conditions emerged naturally in my research:

  1. For all $a\in A$ and $b_1,b_2\in B$ such that $f(a)=b_1.b_2$ there are $a_1,a_2\in A$ such that $f(a_1)=b_1$, $f(a_2)=b_2$ and $a=a_1.a_2$.
  2. If $f(a)=1$, then $a=1$.

Let me point out one elementary example to motivate the whole thing.

Example Let $K$ be a field, let $K_m$ be the multiplicative monoid of monic polynomials over $K$. Then the mapping $d\colon K_m\to\mathbb{N}$ that takes a polynomial to its degree is a morphism of monoids (here, $\mathbb N$ is equipped with $+$, of course).

Now, $d$ satisfies the conditions 1. and 2. if and only if $K$ is algebraically closed.

There are several examples similar to that one: roughly speaking, $A$ contains some things, $B$ is commutative and $f$ means something like "measure/dimension/size" of the things from $A$ valued in $B$. If we allow $A$ to be a partial monoid, the number of interesting examples increases considerably.

Question 1 Was condition 1. used by somebody in the past, within the context of monoids (or semigroups) ?

Question 2 (soft version of Q1) Has anyone seen anything like this before?

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Assume that instead of 2. we assume:

2'. If $f(a)=b_1 b_2$ has two preimage factorizations $a=a_1 a_2 = a'_1 a'_2$, then there is some element $u \in A$ such that $f(u)=1$, $u a'_2 = a_2$ and $a_1 u = a'_1$.

Then this is the notion of a Conduché functor applied to categories with one object.

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