The Lyapin's Notebook — Problem 16

Definitions and mathematical statement

For a semigroup $S$, multiplication is associative: $(xy)z=x(yz)$ for all $x,y,z\in S$. An endomorphism of $S$ is a map $f:S\to S$ satisfying $f(xy)=f(x)f(y)$ for all $x,y\in S$; write $\operatorname{End}(S)$ for the set of such maps.

Source problem (S. Kublanovsky). Obtain estimates for the values of the "endomorphic function" for various basic classes of semigroups.

Unresolved formulation. The supplied record does not define the term "endomorphic function," specify its arguments or codomain, identify the intended basic classes of semigroups, or state the parameters and type of estimates sought. In particular, the definition of an endomorphism above does not determine the intended function, and no identification of that function with $\lvert\operatorname{End}(S)\rvert$ is asserted. The supplied formulation therefore cannot be converted into a precise quantified estimation problem without additional source information.

Short English statement

The problem, attributed to S. Kublanovsky, asks for estimates of an unspecified "endomorphic function" on unspecified basic classes of semigroups. Its precise mathematical target remains unresolved because the function and the intended classes are not defined in the supplied record.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800016$, exact source code catalogue entry, title The Lyapin's Notebook --- Problem 16, in the AMR Open Problem Lists collection. The registry labels the problem open; this is a supplied status label, not an independently verified mathematical or literature assessment.
• Bershadsky, Kublanovsky, Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 16. Supplied source URL: https://arxiv.org/abs/2604.04763. The background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent access or verification.
• The problem statement attributes the question to S. Kublanovsky. The registry's proposer field lists Bershadsky, Kublanovsky, Mashevitzky and its proposed-year field gives 2026; these collection-level entries do not establish the original date of Kublanovsky's question.