The Lyapin's Notebook — Problem 40

Definitions and mathematical statement

A semigroup is a set equipped with an associative binary operation. A semigroup variety is a class of semigroups defined by identities. The supplied problem uses the term inclusive variety without defining it; consequently, the following formulation is conditional on recovering that term's intended meaning from the source.

For an inclusive semigroup variety $\mathcal V$, write \[ \mathcal I(\mathcal V)=\{\mathcal W\subseteq\mathcal V: \mathcal W\text{ is an inclusive semigroup variety}\}, \] ordered by inclusion. The source calls this ordered collection the lattice of inclusive subvarieties; its lattice structure cannot be specified without the missing definition of inclusiveness.

Problem (G. Mashevitzky). Determine all inclusive varieties $\mathcal V$ for which \[ |\mathcal I(\mathcal V)|<\aleph_0 \qquad\text{or}\qquad |\mathcal I(\mathcal V)|=\aleph_0. \] Thus the requested classification includes both finite and countably infinite lattices of inclusive subvarieties. The source reports a partial solution in reference [M2], whose bibliographic entry and mathematical content are absent from the supplied record.

Short English statement

Classify the inclusive varieties whose lattices of inclusive subvarieties are finite or countably infinite. The meaning of "inclusive" must be recovered before this becomes a fully specified mathematical problem. The supplied record labels the problem open and reports a partial solution in [M2]; neither assessment is independently verified here.

Sources

• Local UnsolvedMath registry, permanent ID $800038$, source code catalogue entry, title The Lyapin's Notebook --- Problem 40. The supplied registry records status "open" and proposed year 2026; these are source metadata rather than independently verified historical or mathematical conclusions.
• Bershadsky, Kublanovsky, and Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 40. 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 a claim of access or verification here.
• The problem statement attributes the question to G. Mashevitzky. The registry's proposed_by field lists Bershadsky, Kublanovsky, and Mashevitzky.
• Reference [M2] is cited in the supplied statement as containing a partial solution. Its bibliographic entry is not supplied, so its identity, scope, and conclusions remain unresolved.