About semimodules over the trivial semiring

E. M. Vechtomov, A. A. Petrov

Vyatka State University (Kirov)

Abstract: The article studies semimodules over a single-element semiring $\{e\}$, which we call a trivial semiring. By a semimodule over a trivial semiring we mean a commutative semigroup $\left\langle A, +\right\rangle$ considered together with the mapping $e: A \to A$, $a\to ea$ which is additive, i. e. $e(a+b)=ea+eb$ for any $a, b\in A$; and idempotent, i. e. $e(ea)=ea$ for all $a\in A$; $ea+ea=ea$ for any $a\in A$. In this case, the mapping $e: A \to A$, or the action of $e$ onto $A$, is called a retraction of the commutative semigroup $\left\langle A, +\right\rangle$. For the retraction of $e$ onto $A$, the set $eA$ will be the set of all fixed points of the mapping $e$, called the $e$-set. A commutative semigroup $\left\langle A, +\right\rangle$ can have very different retractions and, accordingly, different $e$-sets. Moreover, the same set on a semilattice $A$ can serve as an $e$-set of different retractions of $e$ onto $A$.
The article shows, using a number of examples, that it is advisable to study retractions on semilattices $\left\langle A, +\right\rangle$, which we call e-semimodules.
The paper provides some classification of retractions, describes the structure of chain retractions. It is proved that all non-empty subsets of an arbitrary chain are $e$-sets if and only if this chain is discrete. We considered increasing, decreasing and linear retractions on semilattices and lattices. It is shown that increasing retractions $e$ and decreasing retractions $e$ are uniquely determined by their $e$-sets.
We also obtained other results, and gave corresponding examples in the paper.

Keywords: semiring, semimodule, retraction, trivial semiring, semilattice, $e$-semimodule, $e$-set, lattice, chane.

UDC: 512.558

Received: 11.01.2025
Revised: 27.08.2025

DOI: 10.22405/2226-8383-2025-26-3-71-80