Extremal problems in combinatorial semigroup theory

Abstract

In this thesis we shall consider three types of extremal problems (i.e. problems involving maxima and minima) concerning semigroups. In the first chapter we show how to construct a minimal semigroup presentation that defines a group of non-negative deficiency given a minimal group presentation for that group. This demonstrates that the semigroup deficiency of a group of non-negative deficiency is equal to the group deficiency of that group. Given a finite monoid we find a necessary and sufficient condition for the monoid deficiency to equal the semigroup deficiency. We give a class of infinite monoids for which this equality also holds. The second type of problem we consider concerns infinite semigroups of relations and transformations. We find the relative rank of the full transformation semigroup, over an infinite set, modulo some standard subsets and subsemigroups, including the set of contraction maps and the set of order preserving maps (for some infinite ordered sets). We also find the relative rank of the semigroup of all binary relations (over an infinite set) modulo the partial transformation semigroup, the full transformation semigroup, the symmetric inverse semigroup, the symmetric group and the set of idempotent relations. Analogous results are also proven for the symmetric inverse semigroup. The third, and final, type of problem studied concerns generalising notions of independence from linear algebra to semigroups and groups. We determine the maximum cardinality of an independent set in finite abelian groups, Brandt semigroups, free nilpotent semigroups, and some examples of infinite groups.