On the subsemigroup complex of an aperiodic Brandt semigroup
- 48 Downloads
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.
KeywordsBrandt semigroup Lattice of subsemigroups Simplicial complex Boolean representable simplicial complex Matroid
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.
- 4.Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. 1. Mathematical Surveys of the American Mathematical Society, No. 7, Providence (1961)Google Scholar
- 8.Izhakian, Z., Rhodes, J.: New Representations of Matroids and Generalizations, preprint, arXiv:1103.0503 (2011)
- 9.Izhakian, Z., Rhodes, J.: Boolean Representations of Matroids and Lattices, preprint, arXiv:1108.1473 (2011)
- 10.Izhakian, Z., Rhodes, J.: C-Independence and c-Rank of Posets and Lattices, preprint, arXiv:1110.3553 (2011)
- 17.Margolis, S., Rhodes, J., Silva, P.V.: On the Dowling and Rhodes Lattices and Wreath Products, arxiv: 1710.05314, preprint (2017)
- 19.Pfeiffer, G.: Counting transitive relations. J. Integer Seq. 7, Article 0.4.32 (2004)Google Scholar
- 20.Rhodes, J., Silva, P.V.: Boolean Representations of Simplicial Complexes and Matroids, Springer Monographs in Mathematics. Springer, Berlin (2015)Google Scholar
- 21.Rhodes, J., Steinberg, B.: The q-Theory of Finite Semigroups, Springer Monographs in Mathematics (2009)Google Scholar