The lattice of semigroups of the semigroup of all mappings on an infinite set

Friday 5th July, 14:00-15:30, UWS, Parramatta campus, Building EB, Room 1.32

Speaker:

James Mitchell (University of St Andrews)

Title:

THE LATTICE OF SUBSEMIGROUPS OF THE SEMIGROUP OF ALL MAPPINGS ON AN INFINITE SET

Abstract:

In this talk I will review some recent results relating to the lattice of subgroups of the symmetric group and its semigroup theoretic counterpart, the lattice of subsemigroups of the full transformation semigroup on an infinite set. As might be expected, these lattices are extremely complicated.

I will discuss several results that make this comment more precise, and shed light on the maximal proper sub(semi)groups in the lattice. I will also discuss a natural related partial order, introduced by Bergman and Shelah, which is obtained by restricting the type of sub(semi)groups and considering classes of, rather than individual, (semi)groups. In the case of the symmetric group, this order is very simple but in the case of the full transformation semigroup it is again very complex.

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