On a variety of commutative multiplicatively idempotent semirings

TL;DR

This paper investigates a specific class of commutative semirings with idempotent multiplication, providing structural insights, a normal form system, and solvability of the word problem, revealing both finiteness and complexity properties.

Contribution

It characterizes the variety generated by these semirings, introduces a normal form system, and proves the word problem is solvable within this class.

Findings

The variety is generated by single semirings.

A normal form system for terms is established.

The word problem in the variety is solvable.

Abstract

We prove that the variety V of commutative multiplicatively idempotent semirings satisfying x + y + xyz = x + y is generated by single semirings. Moreover, we describe a normal form system for terms in V and we show that the word problem in V is solvable. Although V is locally finite, it is residually big.