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====== Speaker: Casey Donoven (Binghamton) ======

====== Title: Intersection Numbers and Minimal Subbases ======

===== Combinatorics Seminar,  =====

Given a finite semigroup $S$, define $\Phi(S)$ to be the intersection of all maximal subsemigroups of $S$, also known as the Frattini subsemigroup of $S$.  The intersection number of $S$ is the minimum number of maximal subsemigroups whose intersection is $\Phi(S)$.  I will speak about a particular example of a semigroup with an interesting intersection number.  In this example, the intersection number is equivalent to the minimum size of a subbasis of the discrete topology on a finite set, which is known.