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.
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Acknowledgements Stuart Margolis acknowledges support from the Binational Science Foundation (BSF) of the United States and Israel, Grant Number 2012080. John Rhodes acknowledges support from the Simons Foundation. Pedro V. Silva was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds (FEDER), under the partnership agreement PT2020.
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- Boolean representable simplicial complex
- Brandt semigroup
- Lattice of subsemigroups
- Simplicial complex