The Lyapin's Notebook — Problem 30

Definitions and mathematical statement

A semigroup is a nonempty set $S$ equipped with an associative binary operation. It is regular if \[ \forall a\in S\;\exists x\in S:\quad axa=a. \] It is aperiodic if every subgroup of $S$ is trivial. Here a subgroup may have an identity different from an identity of the whole semigroup, and trivial means having exactly one element. Let \[ \mathcal{K}=\{S: S\text{ is a finite regular aperiodic semigroup}\}. \] Problem (S. I. Kublanovsky). Is the class $\mathcal{K}$ completely solvable?

The supplied record does not define what it means for a class of semigroups to be completely solvable. The question retains that source terminology without assigning it an interpretation. A precise definition of this property is required to obtain a fully specified mathematical target.

Short English statement

S. I. Kublanovsky asks whether the class of all finite regular semigroups whose subgroups are trivial is completely solvable. The supplied record leaves the meaning of "completely solvable" unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $800029$, source code catalogue entry, title "The Lyapin's Notebook --- Problem 30", upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry labels the problem open; this is supplied source metadata, not an independently verified status assessment.
• Bershadsky, Kublanovsky, Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 30. Supplied source URL: https://arxiv.org/abs/2604.04763. The problem text attributes this question to S. I. Kublanovsky; the registry's proposer field lists Bershadsky, Kublanovsky, Mashevitzky and gives the year 2026.
• The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; the source has not been independently consulted for this draft.