The Lyapin's Notebook — Problem 47

Definitions and mathematical statement

A semigroup is a nonempty set equipped with an associative binary operation. A semigroup variety is a class of semigroups defined by identities in this binary operation. For a semigroup $S$, let $\operatorname{var}(S)$ denote the smallest semigroup variety containing $S$. A subvariety of a variety $\mathcal V$ is a semigroup variety $\mathcal W\subseteq\mathcal V$.

The supplied record does not define the term inclusive variety. Consequently, the following is a conditional formalization with that term left unresolved: let $\mathfrak I$ denote the class of semigroup varieties called inclusive in the source, and put \[ \mathcal L_{\mathfrak I}(\mathcal V) =\{\mathcal W:\mathcal W\text{ is a semigroup variety},\ \mathcal W\subseteq\mathcal V,\ \mathcal W\in\mathfrak I\}, \] ordered by inclusion. The source calls this ordered collection the lattice of inclusive subvarieties; its lattice structure cannot be specified further without the missing definition.

Question (G. Mashevitzky; source Problem 47). Does there exist a finite semigroup $S$ such that \[ \operatorname{var}(S)\in\mathfrak I \qquad\text{and}\qquad \mathcal L_{\mathfrak I}(\operatorname{var}(S))\text{ is infinite}? \] The intended class $\mathfrak I$ must be identified before this becomes a fully specified mathematical question.

The supplied background asserts that if an inclusive variety is generated by a finite group or by a finite nilsemigroup, then its lattice of inclusive subvarieties is finite, citing $[\mathrm{M3}]$. Here a group is regarded as a semigroup under its multiplication, and a nilsemigroup means a semigroup with an absorbing zero $0$ such that for every element $x$ there is an integer $n\geq 1$ with $x^n=0$. This background assertion is reported from the source and has not been independently verified; the bibliographic identity of $[\mathrm{M3}]$ is not supplied.

Short English statement

G. Mashevitzky asks whether a finite semigroup can generate an inclusive variety having infinitely many inclusive subvarieties. The record reports finiteness for inclusive varieties generated by finite groups or finite nilsemigroups. A complete formulation requires the missing definition of inclusive variety. The registry labels the problem open; that status is supplied metadata.

Sources

• Local UnsolvedMath registry, permanent ID $800045$, exact source code catalogue entry, title The Lyapin's Notebook --- Problem 47. The supplied record labels the problem open and attributes the question itself to G. Mashevitzky.
• Bershadsky, Kublanovsky, and Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 47. 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 an independent source visit or verification.
• Reference $[\mathrm{M3}]$, cited in the supplied statement for the finiteness assertion concerning finite groups and finite nilsemigroups. No bibliographic details or link for this reference are supplied.