On the subsemigroup complex of an aperiodic Brandt semigroup

Taking as departure point an article by Cameron, Gadouleau, Mitchell and Peresse on maximal
lengths of subsemigroup chains, we introduce the subsemigroup complex H(S) of a
nite semigroup
S as a (boolean representable) simplicial complex de
fined through chains in the lattice of subsemi-
groups of S. The rank of H(S) is the above maximal length minus one and H(S) provides other
useful invariants concerning 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 independence and bases, asymptotical estimates on
the number of bases, or establishing when the complex is pure or a matroid.

This is joint work with Stuart Margolis (Bar-Ilan University, Ramat Gan, Israel) and John
Rhodes (University of California, Berkeley, USA).