The solution of an open problem on semigroup inclusion classes

TL;DR

This paper characterizes a specific semigroup inclusion class to answer a previously open question about its relation to a larger class, showing it is not covered by that class.

Contribution

It provides a detailed characterization of the semigroup inclusion class $ extbf{I}$, resolving an open problem posed by Monzo regarding its coverage.

Findings

$ extbf{I}$ is not covered by $ extbf{GRB}$.

The paper offers a characterization of $ extbf{I}$ that simplifies understanding its structure.

Answers Monzo's question in the negative.

Abstract

The semigroup inclusion class $\mathbf{I} = [xyxy = xy; xyz \in \{xywz, xuyz\}]$ is the union of two maximal subvarieties of $\mathbf{GRB} = [xyzxy=xy]$. Monzo ( arXiv:1411.4860 ) described the lattice of semigroup inclusion classes below $\mathbf{I}$ and asked if $\mathbf{I}$ is covered by $\mathbf{GRB}$. Our main result is a characterization of $\mathbf{I}$ which makes it easy to answer Monzo's question in the negative.