Commutative Multiplicatively Idempotent Semirings with the Identity $x+2xy=x$

E.M. Vechtomov

Vyatka State University, Kirov

Abstract: Semirings with commutative idempotent multiplication in which the identity
$$x+2xy=x$$
holds are studied. Such semirings form a variety of semirings containing all Boolean rings and all distributive lattices. Various characterizations of these semirings are obtained and the lattices of congruences on them are considered. Examples are given and explaining comments are made.

Keywords: multiplicatively idempotent semiring, commutativity, identity $x+2xy=x$, distributive lattice, Boolean ring, lattice of ideals, lattice of congruences.

UDC: 512.558

MSC: 06D05, 08A05

Received: 16.08.2024
Revised: 17.02.2025

DOI: 10.4213/mzm14479

 English version: Mathematical Notes, 2025, 118:2, 246–258

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