Flat extensions of groups and limit varieties of ai-semirings

TL;DR

This paper studies limit varieties of additively idempotent semirings, providing concrete examples, characterizations, and demonstrating the existence of diverse limit varieties with various properties.

Contribution

It introduces new constructions of limit varieties of additively idempotent semirings, including those generated by flat extensions of finite groups and other examples with unique features.

Findings

Constructed an infinite family of limit additively idempotent semiring varieties.

Identified examples generated by finite flat semirings and characterized limit varieties from flat extensions of finite groups.

Proved the existence of continuum-sized families of limit varieties with no finite generator.

Abstract

The present paper is a continuation of \cite{jrz} and is devoted to the study of limit varieties of additively idempotent semirings. A limit variety is a nonfinitely based variety whose proper subvarieties are all finitely based. We present concrete constructions for one infinite family of limit additively idempotent semiring varieties, and one further ad hoc example. Each of these examples can be generated by a finite flat semiring, with the infinite family arising by a way of a complete characterisation of limit varieties that can be generated by the flat extension of a finite group. We also demonstrate the existence of other examples of limit varieties of additively idempotent semirings, including one further continuum-sized family, each with no finite generator, and two further ad hoc examples. While an explicit description of these latter examples is not given, one of the examples is proved to contain only trivial flat semirings.