Simple semirings with a bi-absorbing element
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Simple semirings with a bi‑absorbing element Tomáš Kepka1 · Miroslav Korbelář2 · Petr Němec3 Received: 11 August 2019 / Accepted: 16 March 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study additively idempotent congruence-simple semirings with a bi-absorbing element. We characterize a subclass of these semirings in terms of semimodules of a special type (o-characteristic semimodules). We show that o-characteristic semimodules are uniquely determined. We also generalize a result by Ježek and Kepka on simple semirings of endomorphisms of semilattices. Keywords Simple semiring · Bi-absorbing · Semimodule · Idempotent · Semilattice Simple semirings are the structural keystones of semirings. A complete classification of commutative simple semirings was obtained in [1]. Finite simple semirings were classified in [6] with the exception of the case of additively idempotent semirings with a bi-absorbing element. In this paper we provide results on additively idempotent semirings with a biabsorbing element that will generalize the finite types studied in [6]. Our results will include also infinite cases. We give a characterization of a subclass of these semirings in terms of o-characteristic semimodules. Our main result (Theorem 2.2) Communicated by Laszlo Marki. The second author was supported by the project CAAS CZ.02.1.01/0.0/0.0/16_019/0000778. * Miroslav Korbelář [email protected] Tomáš Kepka [email protected] Petr Němec [email protected] 1
Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
2
Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27 Prague 6, Czech Republic
3
Department of Mathematics, Czech University of Life Sciences in Prague, Kamýcká 129, 165 21 Suchdol, Prague 6, Czech Republic
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will be an analogy of a similar characterization achieved for the case of additively idempotent semirings with a zero [7, Theorem 5.1]. We also provide a generalization (Theorem 2.4) of a result on endomorphisms of semilattices [5, Theorem 2.2].
1 Preliminaries A semiring S = S(+, ⋅) is an algebraic structure equipped with two associative operations, where the addition is commutative and the multiplication distributes over the addition from both sides. The semiring S is called (congruence-)simple if it has precisely two congruences. A (left) S-semimodule M = S M is a commutative semigroup M(+) together with a semiring homomorphism 𝜑 ∶ S → End(M(+)) usually denoted as an action of S on M in the form sm ∶= 𝜑(s)(m) for all s ∈ S and m ∈ M . A semimodule S M is called faithful if 𝜑 is injective (i.e., if for all a, b ∈ S , a ≠ b , there is at least one x ∈ M with ax ≠ bx ), simple if S M has precisely two (S-semimodule) congruences and minimal if |M| ≥ 2 and for every subsemimodule S N of S M such that |N| ≥ 2 is N = M . A non-empty subset I of S is a left ideal if SI ∪ (I + I) ⊆ I
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