On Double Inverse Semigroups

Abstract

A double semigroup is a set equipped with two associative binary operations satisfying the middle-four interchange law. A double inverse semigroup is a double semigroup in which both operations are inverse semigroup operations. It is shown by Kock (2007) that all double inverse semigroups must be commutative. In this thesis, we define the notion of a double inductive groupoid which admits both a construction of double inverse semigroups given any double inductive groupoid, and vice-versa. These constructions are functorial and induce an isomorphism of categories between the category of double inductive groupoids with inductive functors and double inverse semigroups with double semigroup homomorphisms. By a further investigation of double inverse semigroups, we are able to show that the two operations of any double inverse semigroups must coincide and thus double inverse semigroups are commutative inverse semigroups.