Thin MC left regular bands
Aram Dermenjian
TL;DR
The paper studies MC left regular bands, a class of left regular bands whose face posets are meet-semilattices and whose adjacency graphs on chambers are connected. It introduces an edge-labeled adjacency graph framework, uses the support lattice to label adjacencies, and proves that for finite thin MC LRBs every simple cycle has an even number of edges labeled by equal-support facets, with each such label appearing exactly once; this leads to a bijection between thin LRB graphs and rank-2 thin MC LRBs. A constructive approach then realizes rank-2 thin MC LRBs from thin LRB graphs via explicit multiplication rules, establishing a tight correspondence between the graph-theoretic and algebraic structures. The work suggests further directions, including generalizing beyond thin and rank-2 cases, exploring higher-rank simplicial variants, and connecting to broader combinatorial geometries such as hyperplane arrangements and oriented matroids. Overall, the framework provides concrete combinatorial tools to analyze the adjacency structure of left regular bands and sets up a pathway to classify low-rank instances via labeled graphs with parity constraints on cycles.
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.