Commutative semigroups with cancellative endomorphism semigroups

Definitions and mathematical statement

A semigroup is a nonempty set with an associative multiplication. For a commutative semigroup $S$, let \[ E(S)=\{f:S\longrightarrow S:f(xy)=f(x)f(y)\text{ for all }x,y\in S\}. \] Give this set the pointwise multiplication $(fg)(x)=f(x)g(x)$. This is again a commutative semigroup.

Classify the commutative semigroups $S$ for which $E(S)$ is cancellative: \[ f,g,h\in E(S),\quad fh=gh\quad\Longrightarrow\quad f=g. \] There is no finite-generation assumption on $S$. Endomorphisms mean semigroup homomorphisms; an identity element, if present, need not be preserved.

Short English statement

Which commutative semigroups have cancellation among their endomorphisms when endomorphisms are multiplied point by point?

Sources

A. A. Borisov, Problem 3.8 in S. Bershadsky, S. Kublanovsky and G. Mashevitzky (editors), The Lyapin's Notebook: a collection of unsolved problems in Semigroup Theory, arXiv:2604.04763v1 (6 April 2026), https://arxiv.org/html/2604.04763v1. Adapted with expanded definitions under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/.