The Lyapin's Notebook — Problem 52

Definitions and mathematical statement

Source-attributed problem (E. S. Lyapin). Characterize the pargoids of transformations that admit an internal semigroup extension; that is, give necessary and sufficient conditions for the existence of such an extension, using the meanings of these terms in Problem 52 of the supplied source.

Unresolved definitions and scope. The supplied record does not define a pargoid of transformations or an internal semigroup extension. It does not specify the underlying transformation domains, the partial multiplication involved, or the structural requirements imposed on the extension. Consequently, an explicit quantified formulation cannot be recovered from this record alone without introducing unsupported assumptions. These terms must be resolved from the source before this draft can serve as a precise mathematical statement.

The source adds that some sufficient conditions are given in reference [ESL6]; neither those conditions nor the bibliographic identification of [ESL6] is supplied.

Short English statement

The problem, attributed to E. S. Lyapin, asks for necessary and sufficient conditions under which a pargoid of transformations admits an internal semigroup extension. The essential terminology is undefined in the supplied record, so the formulation remains provisional.

Sources

• Local UnsolvedMath registry, permanent ID $800050$, exact source problem code catalogue entry, title The Lyapin's Notebook --- Problem 52. The supplied registry labels the problem open; this status has not been independently verified.
• Bershadsky, Kublanovsky, Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 52, https://arxiv.org/abs/2604.04763. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not a claim of access or verification for this draft. The problem text attributes the question to E. S. Lyapin.
• Reference [ESL6], cited in the supplied statement for some sufficient conditions. No bibliographic details, link, or statement of these conditions is supplied.