The Lyapin's Notebook — Problem 23

Definitions and mathematical statement

A semigroup is a nonempty set $S$ equipped with an associative binary operation. An element $e\in S$ is idempotent if $e^2=e$. Thus $S$ has no idempotents if $s^2\ne s$ for every $s\in S$. The semigroup $S$ is finitely generated if there is a finite subset $A\subseteq S$ such that every element of $S$ is a nonempty finite product of elements of $A$. It is finitely presented if it is isomorphic to $A^+/\rho$, where $A$ is a finite alphabet, $A^+$ is the free semigroup of nonempty words, and $\rho$ is the congruence generated by a finite set of pairs of words.

Question attributed to A. A. Borisov. Does every finitely generated semigroup without idempotents have finite chain rank? The source also asks the version restricted to finitely presented semigroups. Writing $\operatorname{cr}(S)$ provisionally for the source's chain rank, the intended assertion is \[ \forall S\;\left[\left(S\text{ is a finitely generated semigroup}\right)\land\left(\forall s\in S,\ s^2\ne s\right)\ \Longrightarrow\ \operatorname{cr}(S)<\infty\right], \] with a second version replacing "finitely generated" by "finitely presented." The latter is a restricted version because finite presentation implies finite generation.

Unresolved definition. The supplied record does not define chain rank or identify the chains used to compute it. Consequently, $\operatorname{cr}(S)$ is only a placeholder for the intended invariant, and the displayed assertion is not a complete formal specification until that definition is supplied.

Supplied background qualification. The source reports that finitely generated commutative semigroups without idempotents have finite chain rank, attributing this result to reference [Borisov4]. This is a supplied literature claim, not an independently verified result here.

Short English statement

The question asks whether every finitely generated semigroup with no idempotent elements has finite chain rank, and also asks the restricted question for finitely presented semigroups. The supplied source reports an affirmative result for commutative semigroups. Chain rank is not defined in the record, so the statement requires clarification and human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $800022$, source problem code catalogue entry, title The Lyapin's Notebook --- Problem 23. The registry labels the problem open; this is supplied status metadata, not an independently verified assessment.
• Bershadsky, Kublanovsky, and Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 23. Supplied source URL: https://arxiv.org/abs/2604.04763. The record gives access date 2026-10-16 and extraction method arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.
• The problem text attributes the question to A. A. Borisov. The registry's proposer fields instead list Bershadsky, Kublanovsky, and Mashevitzky and the year 2026; these fields do not establish the original date of Borisov's question.
• Reference [Borisov4], cited in the supplied problem text for the commutative case. No bibliographic details or separate URL for this reference are supplied, so the citation cannot be resolved from this record alone.