The Lyapin's Notebook — Problem 4

Definitions and mathematical statement

For a positive integer $n$, let $[n]=\{1,\ldots,n\}$ and let \[ T(n)=\{f:[n]\to[n]\} \] be the full transformation semigroup, with multiplication given by composition, $(fg)(a)=f(g(a))$. The positive-integer range is the usual convention adopted here; the supplied source does not explicitly specify the range of $n$, including whether an empty underlying set is intended.

Let $X=\{x_1,x_2,\ldots\}$, and let $X^+$ be the free semigroup of finite nonempty words over $X$, with concatenation as multiplication. Each assignment $\phi:X\to T(n)$ extends uniquely to a semigroup homomorphism $\widehat\phi:X^+\to T(n)$. A semigroup identity $u\approx v$, where $u,v\in X^+$, holds in $T(n)$ if \[ \widehat\phi(u)=\widehat\phi(v) \qquad\text{for every assignment }\phi:X\to T(n). \] Thus its equational theory is \[ \operatorname{Id}(T(n))= \left\{(u,v)\in X^+\times X^+: \widehat\phi(u)=\widehat\phi(v) \text{ for every }\phi:X\to T(n)\right\}. \] Problem (E. S. Lyapin). Describe $\operatorname{Id}(T(n))$ for an $n$-element underlying set, with the qualification on the range of $n$ stated above. The source asks for a description of the identities; it does not specify a required form of description or ask explicitly for a finite identity basis. Nonempty words are used because the source specifies semigroup identities.

The supplied source reports that some classes of these identities were described in works labeled [PSSSV] and [Ma3]. Their bibliographic details and the classes covered are absent from the supplied record, so this reported partial progress cannot be made more precise here.

Short English statement

Describe exactly which equations between nonempty words hold under every substitution of transformations of an $n$-element set. The problem is attributed to E. S. Lyapin. The supplied registry labels it open and reports partial descriptions in [PSSSV] and [Ma3]; these are supplied source assessments, not independently verified conclusions.

Sources

• Local UnsolvedMath registry, permanent ID $800004$, source problem code catalogue entry, title The Lyapin's Notebook --- Problem 4; supplied complete source record, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record was created and updated on 2026-10-16 and carries the status open.
• Bershadsky, Kublanovsky, Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 4, https://arxiv.org/abs/2604.04763. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; this access information is supplied provenance, not a claim of independent consultation. The statement attributes the problem to E. S. Lyapin; the registry's proposer field names Bershadsky, Kublanovsky, Mashevitzky and gives the year 2026.
• [PSSSV]: reference label supplied for work reportedly describing some classes of identities of $T(n)$. No bibliographic details or separate link are supplied.
• [Ma3]: reference label supplied for work reportedly describing some classes of identities of $T(n)$. No bibliographic details or separate link are supplied.