The Lyapin's Notebook — Problem 24

Definitions and mathematical statement

A commutative semigroup is a nonempty set $S$ with an associative binary operation $\cdot:S\times S\to S$ satisfying $xy=yx$ for all $x,y\in S$. No identity element or finiteness assumption is imposed. A semigroup endomorphism is a map $f:S\to S$ such that \[ f(xy)=f(x)f(y)\qquad(x,y\in S). \] Write \[ \operatorname{End}(S)=\{f:S\to S: f(xy)=f(x)f(y)\text{ for all }x,y\in S\}. \] Here endomorphisms preserve the semigroup operation; no additional identity-preservation condition is imposed.

Problem (A. A. Borisov; Problem 24 in the supplied source). Describe all commutative semigroups $S$ satisfying \[ \lvert\operatorname{End}(S)\rvert=4. \] The question has no stated restriction to finite semigroups.

The supplied source reports that reference [Borisov3] proves the following finite-case restriction: \[ \left(S\text{ is a finite commutative semigroup and }|S|>4\right) \ \Longrightarrow\ |\operatorname{End}(S)|\geq 5. \] This is a supplied literature claim, not an independently verified result here. The bibliographic identity of [Borisov3] is absent from the supplied record, so its precise hypotheses cannot be checked from this evidence.

Short English statement

Describe all commutative semigroups having exactly four multiplication-preserving self-maps. The problem allows both finite and infinite semigroups. The source attributes the question to A. A. Borisov and reports that a finite commutative semigroup with more than four elements has at least five endomorphisms; the cited reference [Borisov3] is not identified bibliographically in the supplied record.

Sources

• Local UnsolvedMath registry: permanent ID $800023$, source problem code catalogue entry, title The Lyapin's Notebook --- Problem 24; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied registry labels the problem open; this label is not an independently verified mathematical assessment.
• Bershadsky, Kublanovsky, and Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 24. Supplied source URL: https://arxiv.org/abs/2604.04763. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of source access here.
• The statement attributes the problem to A. A. Borisov. The registry's proposed_by field lists Bershadsky, Kublanovsky, and Mashevitzky, and its proposed_year field gives 2026; these registry fields do not establish the original date of Borisov's question.
• Reference [Borisov3] is cited in the supplied statement for the finite-case restriction. No author expansion, title, publication details, or link for this reference is supplied.