The Lyapin's Notebook — Problem 46

Definitions and mathematical statement

A semigroup is a nonempty set equipped with an associative binary operation; an identity element is not required. A variety of semigroups is the class of all semigroups satisfying some fixed set of identities in the language of one binary operation. For a semigroup variety $\mathcal V$, define \[ \mathcal V_{\mathrm{fin}}=\{S\in\mathcal V: |S|<\infty\}, \qquad \mathcal V_{\mathrm{fg}}=\{S\in\mathcal V: S=\langle A\rangle\text{ for some finite nonempty }A\subseteq S\}. \] Here $\langle A\rangle$ consists of all finite nonempty products of elements of $A$.

The supplied question, attributed to G. Mashevitzky, has two indicated versions:

1. Does there exist a semigroup variety $\mathcal V$ such that $\mathcal V_{\mathrm{fin}}$ forms an elementary inclusive variety?
2. Does there exist a semigroup variety $\mathcal V$ such that $\mathcal V_{\mathrm{fg}}$ forms an elementary inclusive variety?

The source uses the wording "finite (or finitely generated)"; the two versions above retain these alternatives without requiring the same variety to satisfy both. The supplied record does not define elementary inclusive variety. Consequently, the target property remains unspecified, and these questions cannot be made fully precise from the supplied evidence alone. In particular, no identification of that phrase with a first-order elementary class or an ordinary equational variety is assumed. No additional nontriviality restriction on $\mathcal V$ is supplied.

Short English statement

G. Mashevitzky asks whether some semigroup variety has its finite members, or alternatively its finitely generated members, forming an "elementary inclusive variety." The supplied record does not define this target property, so its intended meaning requires clarification.

Sources

• Local UnsolvedMath problem registry, permanent ID $800044$, source code catalogue entry, title The Lyapin's Notebook --- Problem 46. The supplied registry labels the problem open; this label is not an independently verified mathematical assessment.
• Bershadsky, Kublanovsky, Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 46. Supplied source URL: https://arxiv.org/abs/2604.04763. The problem statement attributes the question specifically to G. Mashevitzky; the registry's proposer field lists Bershadsky, Kublanovsky, Mashevitzky.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance claims, not an assertion that the source was visited or independently verified for this draft.