The Lyapin's Notebook — Problem 15

Definitions and mathematical statement

A semigroup is a nonempty set $S$ equipped with an associative binary operation $S\times S\to S$. A variety of semigroups is a class of semigroups defined by identities in the language of one binary operation. An endomorphism of a semigroup $S$ is a map $h:S\to S$ satisfying $h(xy)=h(x)h(y)$ for all $x,y\in S$.

The supplied Problem 15, attributed to S. Kublanovsky, asks:

Is it true that for any endomorphic variety of semigroups its endomorphic function is identical?

The quantifier ranges over every variety satisfying the source's unspecified condition of being endomorphic. The supplied record does not define that condition, the associated endomorphic function, or the meaning of identical. In particular, it gives neither the domain nor the codomain of that function. Consequently, the question is retained in its supplied wording; a fully specified mathematical assertion requires these missing definitions. The standard definition of a semigroup endomorphism above does not supply definitions of these additional terms.

Short English statement

The question attributed to S. Kublanovsky asks whether every endomorphic variety of semigroups has an endomorphic function that is "identical." The specialized terms and the intended equality are undefined in the supplied record, so the target requires clarification from the source. The registry labels the problem open; that label is supplied metadata, not an independently verified assessment.

Sources

• Local UnsolvedMath registry, permanent ID $800015$, source problem code catalogue entry, title The Lyapin's Notebook --- Problem 15. The record lists Bershadsky, Kublanovsky, and Mashevitzky under proposer metadata, with year 2026; the statement specifically attributes the question to S. Kublanovsky. The supplied status is open.
• Bershadsky, Kublanovsky, and Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 15. Supplied source URL: https://arxiv.org/abs/2604.04763. The record reports access on 2026-10-16 and extraction method arxiv_latex. These are supplied provenance details; the source and its contents have not been independently checked for this draft.