A finitely based finite semiring generates a variety with continuum many subvarieties

TL;DR

This paper proves the existence of finitely based finite semirings whose varieties have continuum many subvarieties, using homomorphism theory of Kneser graphs, and provides specific examples including the 3-element semiring S_{53}.

Contribution

It introduces the first known example of a finitely based finite semiring with a variety of continuum many subvarieties, expanding understanding of semiring variety complexity.

Findings

The 3-element semiring S_{53} has a variety with continuum many subvarieties.

The max-plus semiring's variety is also of type 2^{}.

The 4-element semiring B_0's variety contains infinitely many subvarieties.

Abstract

This paper establishes the existence of a finitely based finite semiring whose variety contains a continuum of subvarieties; such a variety is said to be of type \(2^{\aleph_0}\). Using the homomorphism theory of Kneser graphs, we prove that the 3-element semiring \(S_{53}\) is the first known example with this property. Moreover, \(S_{53}\) belongs to the variety of the max-plus semiring \((\mathbb{N},\max,+)\), which therefore is also of type \(2^{\aleph_0}\). For the finitely based 4-element semiring \(B_0\), we demonstrate that its variety contains infinitely many subvarieties and suggest that \(B_0\) could be another potential example of type \(2^{\aleph_0}\).