On Moore families in semigroups

The properties of Moore families and dual Moore families are studied in general and several specific occurrences of each in semi-groups are shown. In particular, the family of all regular subsets of a semigroup is shown to be a dual Moore family and, based on this fact, the regular core of a subset of a semigroup is defined. The regular core of a Green's class of a semigroup is described and an example is given to show that a J-class J may have a non-empty regular core that is a proper subset of J. The thesis is concluded with a study of congruence relations on a semigroup by use of the properties of Moore families.