The Lyapin's Notebook — Problem 17

Definitions and mathematical statement

A semigroup is a nonempty set $S$ equipped with an associative binary operation. It is finitely generated if there is a finite nonempty subset $A\subseteq S$ such that every $s\in S$ can be written as $s=a_1\cdots a_n$ for some integer $n\geq 1$ and elements $a_1,\ldots,a_n\in A$. An element $e\in S$ is idempotent if $e^2=e$; thus having no idempotents means \[ \forall e\in S,\qquad e^2\neq e. \] Problem (A. A. Borisov). Describe the finitely generated Lesokhin semigroups having no idempotents. Interpreting "describe" as a request for classification up to semigroup isomorphism, the target is the class \[ \mathcal C=\{S\mid S\text{ is a finitely generated Lesokhin semigroup and }(\forall e\in S)\ e^2\neq e\}. \] Unresolved qualification. The supplied record does not define "Lesokhin semigroup." That term remains an unspecified source-dependent condition here, so the displayed class is not a self-contained mathematical specification. The source also does not specify the desired form of the description.

Supplied background assertion. The record attributes to reference [Borisov4] the assertion that every finitely generated separative Lesokhin semigroup without idempotents is isomorphic to a subsemigroup of an infinite monogenic semigroup. The latter may be represented as \[ M=\{a^n:n\in\mathbb Z_{\geq 1}\},\qquad a^m a^n=a^{m+n}, \] with all powers distinct; equivalently, $M\cong(\mathbb Z_{\geq 1},+)$. Thus the supplied background asserts, under its additional separativity hypothesis, the existence of an injective map $\varphi:S\to\mathbb Z_{\geq 1}$ satisfying \[ \varphi(xy)=\varphi(x)+\varphi(y)\qquad(x,y\in S). \] This is reported as an existing result in the source, not as the requested classification or as an independently verified theorem. Neither the definition of "separative" nor the bibliographic entry [Borisov4] is included in the supplied record; the exact hypotheses and wording of that assertion require review.

Short English statement

The problem, attributed to A. A. Borisov, asks for a description of all finitely generated Lesokhin semigroups with no idempotents. The supplied background reports an embedding result for the additional separative case. A self-contained formulation requires the missing definition of a Lesokhin semigroup and clarification of the intended classification.

Sources

• Local UnsolvedMath registry: permanent ID $800017$, source problem code catalogue entry, title The Lyapin's Notebook --- Problem 17. The supplied registry labels the problem open; this is supplied curation, not an independently verified status assessment.
• Bershadsky, Kublanovsky, Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 17. Supplied source URL: https://arxiv.org/abs/2604.04763. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are source-record provenance claims, not a claim of access or verification for this draft.
• The problem text attributes the question to A. A. Borisov. The registry's proposer field lists Bershadsky, Kublanovsky, Mashevitzky and its proposed-year field gives 2026; the original proposal date is not established by the supplied evidence.
• Reference [Borisov4] is cited in the supplied text for the separative-case assertion. Its author details, title, publication information, and URL are not supplied, so the reference and its exact result cannot be checked from this record alone.