#### Date of Award

Spring 1992

#### Document Type

Dissertation - Restricted

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics, Statistics and Computer Science

#### First Advisor

Pastijn, Francis

#### Abstract

If V is a given variety and F: A-> F(A) a function which assigns "formula", then the F-closed subvarieties of V forms a complete lattice under inclusion. The question arises when this lattice forms a complete sublattice of the lattice of all subvarieties of V, and which such sublattices arise in this way. Further, the structural properties of this lattice can be investigated for any given F. If W is any subvariety of V, then there exists a least F-closed subvariety FW of V which is F-closed and which contains W. The transformation W -> FW is a complete join homomorphism of the lattice of subvarieties of V onto the lattice of F-closed subvarieties of V. The nature of this transformation may be investigated for any given F. Questions of this kind will be considered for the cases where the above mentioned variety V is the variety of all semigroups and for several concrete cases of a F: A-> F(A) which assigns to A an extension F(A) of A which is obtained in some canonical way.