Problem of the Week
Hilton Memorial Lecture
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.