The Bergman property for semigroups

In this talk, we will discuss the Bergman property for semigroups and the associated notions of cofinality and strong confinality. An uncountably infinite semigroup S is said to have the Bergman property if for all generating sets U there exists a number n so that every element of S is a product of at most n elements of U. We will see that the semigroup of all mappings on an infinite set has the Bergman property but that its finitary power semigroup does not; the symmetric inverse semigroup on an infinite set and its finitary power semigroup have the Bergman property; the Baer-Levi semigroup does not have the Bergman property.