The Lyapin's Notebook — Problem 19

Definitions and mathematical statement

A semigroup is a nonempty set with an associative multiplication. It is commutative if $xy=yx$ for all elements $x,y$. It is finitely generated if there is a finite subset $A$ such that every element is a product of a positive finite number of elements of $A$. A commutative semigroup $S$ is separative if \[ \forall x,y\in S,\qquad x^2=xy=y^2\ \Longrightarrow\ x=y. \] Problem (A. A. Borisov). Describe the finitely generated commutative separative $E$-semigroups.

Unresolved qualification. The supplied record does not define an $E$-semigroup. Thus the additional property denoted by $E$ must be recovered from the source before this classification request has a fully specified mathematical domain. No definition of that property is assumed here, and the accompanying endomorphism statement alone does not determine it.

Supplied background assertion. For a commutative semigroup $S$, let $\operatorname{End}(S)$ be its set of semigroup endomorphisms, equipped with pointwise multiplication \[ (fg)(s)=f(s)g(s)\qquad(f,g\in\operatorname{End}(S),\ s\in S). \] This multiplication makes $\operatorname{End}(S)$ a commutative semigroup. Under the usual convention, a commutative semigroup $T$ is Archimedean if \[ \forall a,b\in T\ \exists n\in{\mathbb{Z}}_{\geq 1}\ \exists u\in T^1:\quad a^n=bu, \] where $T^1$ denotes $T$ with an identity adjoined when necessary. The source reports, with reference label $[\mathrm{Borisov4}]$, that for a finitely defined commutative semigroup it is algorithmically decidable whether its endomorphism semigroup under pointwise multiplication is Archimedean. Here "finitely defined" is read in its usual sense of admitting a finite presentation by generators and relations. This reported decidability assertion is background, not a resolution of the classification request.

Short English statement

The problem, attributed to A. A. Borisov, asks for a description of finitely generated commutative separative $E$-semigroups. The meaning of $E$-semigroup is missing from the supplied record, so the target class remains incompletely specified. The source also reports a decidability result about Archimedean endomorphism semigroups for finitely presented commutative semigroups.

Sources

• Local UnsolvedMath registry, permanent ID $800019$, source code catalogue entry, title "The Lyapin's Notebook --- Problem 19." The supplied registry labels the problem open; this status has not been independently verified.
• Bershadsky, Kublanovsky, and Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 19. Supplied source URL: https://arxiv.org/abs/2604.04763. The problem text attributes the question to A. A. Borisov; the registry's proposer fields name the three compilation authors and give the year 2026.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details, not a claim of independent source consultation or verification.
• Reference label $[\mathrm{Borisov4}]$ is supplied for the background decidability assertion. Its bibliographic entry is absent from the record, so that citation cannot be identified precisely here.