The Lyapin's Notebook — Problem 21

Definitions and mathematical statement

A semigroup is a nonempty set with an associative binary operation. Let $S$ be a commutative semigroup, written multiplicatively, and define \[ \operatorname{End}(S)=\{f:S\to S: f(xy)=f(x)f(y)\text{ for all }x,y\in S\}. \] Equip this set with pointwise multiplication: \[ (f\star g)(x)=f(x)g(x)\qquad(f,g\in\operatorname{End}(S),\ x\in S). \] Commutativity of $S$ ensures that $f\star g$ is again an endomorphism. Thus $(\operatorname{End}(S),\star)$ is a commutative semigroup. Endomorphisms here are semigroup homomorphisms; if $S$ has an identity, no additional identity-preservation requirement is imposed.

Put \[ {\mathbb{Q}}_{>0}=\{q\in{\mathbb{Q}}:q>0\},\qquad {\mathbb{Q}}_{\geq0}=\{q\in{\mathbb{Q}}:q\geq0\}. \] Problem (A. A. Borisov). Describe, up to semigroup isomorphism, all commutative semigroups $S$ satisfying each of the following conditions separately:

1. $(\operatorname{End}(S),\star)\cong({\mathbb{Q}}_{>0},+)$;
2. $(\operatorname{End}(S),\star)\cong({\mathbb{Q}}_{\geq0},+)$.

Here each isomorphism is a bijection preserving the indicated semigroup operations. The operation on endomorphisms is explicitly pointwise multiplication. No finiteness, cancellation, or identity hypothesis on $S$ is supplied.

Short English statement

Classify the commutative semigroups whose endomorphisms, multiplied pointwise, form a semigroup isomorphic to the positive rationals under addition, and separately classify those giving the nonnegative rationals under addition. The registry labels the problem open; this is a supplied status, not an independently verified assessment.

Sources

• Local UnsolvedMath registry, permanent ID $800021$, source code catalogue entry, title The Lyapin's Notebook --- Problem 21; supplied complete source record, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Bershadsky, Kublanovsky, Mashevitzky, The Lyapin's Notebook: Unsolved Problems in Semigroup Theory (2026), Problem 21. Supplied source URL: https://arxiv.org/abs/2604.04763. The record reports access on 2026-10-16 and extraction by arxiv_latex; these provenance details are supplied and were not independently verified.
• The problem statement attributes the problem to A. A. Borisov. The registry's proposed_by field instead lists Bershadsky, Kublanovsky, Mashevitzky, with proposed_year equal to 2026.