On the variety generated by all semirings of order two

TL;DR

This paper thoroughly analyzes the variety generated by all ten two-element semirings, establishing that it contains exactly 480 subvarieties, each with a finite basis, thus advancing the understanding of semiring algebraic structures.

Contribution

It provides a complete classification of the subvarieties generated by all ten two-element semirings and proves that each is finitely based, extending prior partial results.

Findings

The variety generated by all ten two-element semirings has exactly 480 subvarieties.

Each subvariety within this variety is finitely based.

The work generalizes previous results on specific subclasses of two-element semirings.

Abstract

There are ten distinct two-element semirings up to isomorphism, denoted \( L_2, R_2, M_2, D_2, N_2, T_2, Z_2, W_2, Z_7 \), and \( Z_8 \) (see \cite{bk}). Among these, the multiplicative reductions of \( M_2, D_2, W_2 \), and \( Z_8 \) form semilattices, while the additive reductions of \( L_2, R_2, M_2, D_2, N_2 \), and \( T_2 \) are idempotent semilattices, commonly referred to as \emph{idempotent semirings}. In 2015, Vechtomov and Petrov \cite{vp} studied the variety generated by \( M_2, D_2, W_2 \), and \( Z_8 \), proving that it is finitely based. In the same year, Shao and Ren \cite{srii} examined the variety generated by the six idempotent semirings, demonstrating that every subvariety of this variety is finitely based. This paper systematically investigates the variety generated by all ten two-element semirings. We prove that this variety contains exactly 480 subvarieties, each of which is finitely based.