Thin MC left regular bands

TL;DR

This paper introduces thin MC left regular bands, explores their adjacency graphs, and characterizes their structure using edge labeled graphs with even cycles, providing new insights into their combinatorial properties.

Contribution

It defines thin MC left regular bands, characterizes their adjacency graphs, and introduces thin LRB graphs to encode rank two cases, along with a face poset criterion.

Findings

Adjacency graphs of thin MC left regular bands are represented by edge labeled graphs with even cycles.

Thin LRB graphs encode rank two thin MC left regular bands.

A criterion for when the face poset of a left regular band is a meet-semilattice is provided.

Abstract

We define MC left regular bands and study their adjacency graphs. We prove that for thin MC left regular bands, the adjacency graph is particularly nice and is represented by edge labeled graphs where every simple cycle has an even number of edges. Conversely, we define a set of graphs which we call thin LRB graphs which encode rank two thin MC left regular bands. Along the way, we provide a criterion for showing when the face poset of a left regular band is a meet-semilattice.