Subdirectly irreducible commutative multiplicatively idempotent semirings

Ivan Chajda,Helmut Länger

doi:10.1007/s00012-016-0403-2

Abstract

Commutative multiplicatively idempotent semirings were studied by the authors and F. Svrcek, where the connections to distributive lattices and unitary Boolean rings were established. The variety of these semirings has nice algebraic properties and hence there arose the question to describe this variety, possibly by its subdirectly irreducible members. For the subvariety of so-called Boolean semirings, the subdirectly irreducible members were described by F. Guzman. He showed that there were just two subdirectly irreducible members, which are the 2-element distributive lattice and the 2-element Boolean ring. We are going to show that although commutative multiplicatively idempotent semirings are at first glance a slight modification of Boolean semirings, for each cardinal n > 1, there exist at least two subdirectly irreducible members of cardinality n and at least 2n such members if n is infinite. For \({n \in \{2, 3, 4\}}\) the number of subdirectly irreducible members of cardinality n is exactly 2.

Highlights

Semirings form a useful tool in investigations both in algebra and computer science, see e.g., [4] for details
An important role is played by the variety I of multiplicatively idempotent semirings, since these are close to Boolean rings, which form a base for the classical propositional calculus and are used in computer science, see e.g., [4]
We will show that there exists an infinite number of subdirectly irreducible members that are, linearly ordered, and an infinite number of such members that are not linearly ordered

Summary

Introduction

Semirings form a useful tool in investigations both in algebra and computer science, see e.g., [4] for details. An important role is played by the variety I of multiplicatively idempotent semirings, since these are close to Boolean rings, which form a base for the classical propositional calculus and are used in computer science, see e.g., [4]. In some considerations, it is appropriate to study more general structures than rings and the concept of commutative multiplicatively idempotent semirings is a natural generalization. Our paper is devoted to the problem of determining the subdirectly irreducible members of the variety of commutative members of I. Key words and phrases: semiring, commutative semiring, multiplicatively idempotent semiring, subdirectly irreducible semiring, Boolean semiring. We prove that such algebras form a proper class, i.e., the variety of commutative multiplicatively idempotent semirings is not residually small

Basic notions

Structure of subdirectly irreducible semirings

Constructing subdirectly irreducible semirings

Summary