About semimodules over the trivial semiring

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 ⟨𝐴,+⟩ considered together with the mapping 𝑒 : 𝐴 → 𝐴, 𝑎 → 𝑒𝑎 which is additive, i. e. 𝑒(𝑎+𝑏) = 𝑒𝑎+𝑒𝑏
for any 𝑎, 𝑏 ∈ 𝐴; and idempotent, i. e. 𝑒(𝑒𝑎) = 𝑒𝑎 for all 𝑎 ∈ 𝐴; 𝑒𝑎 + 𝑒𝑎 = 𝑒𝑎 for any 𝑎 ∈ 𝐴.
In this case, the mapping 𝑒 : 𝐴 → 𝐴, or the action of 𝑒 onto 𝐴, is called a retraction of the commutative semigroup ⟨𝐴,+⟩. 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 ⟨𝐴,+⟩ 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 ⟨𝐴,+⟩, 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.