# On the subsemigroup complex of an aperiodic Brandt semigroup

## TL;DR

This paper introduces the subsemigroup complex of finite semigroups, focusing on combinatorial Brandt semigroups, and explores its properties such as face characterization, facet counts, and conditions for purity or matroid structure.

## Contribution

It defines the subsemigroup complex for finite semigroups and provides new characterizations and properties specifically for combinatorial Brandt semigroups.

## Key findings

- Characterizations of faces and facets
- Asymptotic estimates on the number of facets
- Conditions for the complex to be pure or a matroid

## Abstract

We introduce the subsemigroup complex of a finite semigroup S as a (boolean representable) simplicial complex defined through chains in the lattice of subsemigroups of S. We present a research program for such complexes, illustrated through the particular case of combinatorial Brandt semigroups. The results include alternative characterizations of faces and facets, asymptotical estimates on the number of facets, or establishing when the complex is pure or a matroid.