The flat semirings with nilpotent multiplicative reducts

TL;DR

This paper studies a specific class of flat semirings with nilpotent multiplicative structures, characterizing their subvarieties and identifying a unique limit subvariety generated by acyclic graph semirings.

Contribution

Introduces graph semirings to characterize subdirectly irreducible members and analyzes the subvariety structure of the variety _3.

Findings

Uncountably many subvarieties of _3.

Every finitely generated subvariety is a Cross variety.

Existence of a unique limit subvariety generated by acyclic graph semirings.

Abstract

In this paper, we focus on the variety $\mathbf{NF}_3$ generated by all flat semirings with $3$-nilpotent multiplicative reducts. By introducing graph semirings, we characterize all subdirectly irreducible members of $\mathbf{NF}_3$. We prove that the variety $\mathbf{NF}_3$ has uncountably many subvarieties and show that every finitely generated subvariety of $\mathbf{NF}_3$ is a Cross variety. Moreover, we demonstrate that $\mathbf{NF}_3$ has a unique limit subvariety, which is generated by all acyclic graph semirings.