The Lyapin's Notebook — Problem 6

Definitions and mathematical statement

A semigroup is a nonempty set with an associative binary operation. A semigroup variety is a class of semigroups defined by identities. Let $\mathcal L$ be the lattice of all semigroup varieties, ordered by inclusion. For $\mathcal V\in\mathcal L$, define its set of covers by \[ \operatorname{Cov}_{\mathcal L}(\mathcal V)=\{\mathcal W\in\mathcal L:\mathcal V\subsetneq\mathcal W\text{ and there is no }\mathcal U\in\mathcal L\text{ with }\mathcal V\subsetneq\mathcal U\subsetneq\mathcal W\}. \] For a source-intended index $n$ for which the displayed power is meaningful as a semigroup word, put \[ \mathcal E_n=\{S:S\text{ is a semigroup and }a^2=a^{n+2}\text{ for every }a\in S\},\qquad \mathcal L_n=\{\mathcal V\in\mathcal L:\mathcal V\subseteq\mathcal E_n\}. \] Covers in $\mathcal L_n$ are defined by the same strict-inclusion condition, with all intervening varieties required to belong to $\mathcal L_n$.

Problem (S. Kublanovsky). Describe \[ \operatorname{Cov}_{\mathcal L}(RS_n) \quad\text{and}\quad \operatorname{Cov}_{\mathcal L_n}(RS_n). \] The second expression presupposes $RS_n\subseteq\mathcal E_n$. The supplied record does not define the semigroup variety $RS_n$, specify the range of $n$, or state hypotheses establishing this containment. Thus this is a conditional formulation of the requested classification; the target varieties cannot be identified precisely from the supplied record alone.

The source reports that results labeled [Kubl5] imply that $\operatorname{Cov}_{\mathcal L_n}(RS_n)$ is finite. It also reports, citing [Kubl4], that every subvariety of $RS_1$ has finitely many covers. The supplied excerpt does not explicitly specify the ambient lattice for this latter assertion. These are retained as source-reported background statements, without independent verification.

Short English statement

Describe all varieties immediately above $RS_n$, both among all semigroup varieties and among those satisfying $x^2=x^{n+2}$. The supplied record leaves $RS_n$ and the range of $n$ undefined, so the classification target requires clarification. Its finiteness statements are supplied background claims.

Sources

• Local UnsolvedMath registry: permanent ID $800006$, source problem code catalogue entry, title The Lyapin's Notebook --- Problem 6. The registry labels the problem open; this is a supplied status, not an independently verified assessment.
• Bershadsky, Kublanovsky, Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 6, attributed in the statement to S. Kublanovsky. Supplied source URL: https://arxiv.org/abs/2604.04763. The registry records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of access or verification here. The registry's proposer field lists Bershadsky, Kublanovsky, Mashevitzky and its proposed-year field gives 2026.
• [Kubl5]: cited in the supplied statement for finiteness of the covers in the lattice satisfying $x^2=x^{n+2}$. No bibliographic entry or separate URL is supplied.
• [Kubl4]: cited in the supplied statement for finiteness of the covers of every subvariety of $RS_1$. No bibliographic entry or separate URL is supplied.