#### Date of Award

Summer 1994

#### Document Type

Dissertation - Restricted

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics, Statistics and Computer Science

#### First Advisor

Jones, Peter

#### Second Advisor

Bankston, Paul

#### Third Advisor

Byleen, Karl

#### Abstract

It was of great interest to study the relationships between the structure of a group and the structure of its subgroup lattice. Later in the literature the same topic was considered for the case of inverse semigroups. The lattices of inverse subsemigroups and the lattices of full inverse subsemigroups of inverse semigroups were investigated to demonstrate the structure of inverse semigroups. On an inverse semigroup, there exists a natural partial order. The well-known concepts of being closed or being convex can be defined in terms of the partial order. The problem sought in this dissertation is to chacterize inverse semigroups when their lattices of closed [E-closed, or convex] inverse subsemigroups have certain properties, such as distributivity, modularity, semimodularity, or Booleaness, etc. A considerable part of the discussion is attempted to demonstrate how the partial order relates elements. A summary of main results will be found in an appendix.