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<p>We study left regular bands using the adjacency graph of chambers, labeled by facets. In particular, we define a particular family of left regular bands which we call thin MC left regular bands whose face poset is a meet-semilattice and whose adjacency graph is connected. We provide a criterion for when the face poset is a meet-semilattice using the multiplication of the semigroup and its associated support lattice. It turns out that the thin MC left regular bands have a particularly nice adjacency graph. In particular we prove that the adjacency graph of an arbitrary thin MC left regular band is such that each label appears precisely once and every simple cycle has an even number of edges whose associated facets have equal support, implying every simple cycle is of even length. Conversely, we define a set of graphs which we call thin LRB graphs which encode rank two thin MC left regular bands.</p>

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Thin MC left regular bands

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  • 发表日期：

    2025年9月10日

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  • Aram Dermenjian

摘要

We study left regular bands using the adjacency graph of chambers, labeled by facets. In particular, we define a particular family of left regular bands which we call thin MC left regular bands whose face poset is a meet-semilattice and whose adjacency graph is connected. We provide a criterion for when the face poset is a meet-semilattice using the multiplication of the semigroup and its associated support lattice. It turns out that the thin MC left regular bands have a particularly nice adjacency graph. In particular we prove that the adjacency graph of an arbitrary thin MC left regular band is such that each label appears precisely once and every simple cycle has an even number of edges whose associated facets have equal support, implying every simple cycle is of even length. Conversely, we define a set of graphs which we call thin LRB graphs which encode rank two thin MC left regular bands.