## Dissertations (1934 -)

#### Title

Varieties of Pseudosemilattices

Spring 2004

#### Document Type

Dissertation - Restricted

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics, Statistics and Computer Science

Pastijn, Francis

Bankston, Paul

Byleen, Karl

#### Abstract

Due to their large number, we tend to study algebras gathered into classes according to their properties. The concept of variety is a natural way of grouping algebras. Varieties are precisely the classes of algebras defined by identities (for example, groups, semigroups, and rings). However, not all classes of algebras that have been investigated are varieties (for instance, fields). One of those classes is the class of all regular semigroups, which appears naturally when generalizing groups, by generalizing the concept of inverse of an element. Yet, for regular semigroups, there is a weaker concept - e-variety - related to varieties. An important example of an e-variety is the class LI of all locally inverse semigroups. A pseudosemilattice is an idempotent algebra (not a semigroup in general) related to the set of idempotents of a locally inverse semigroup. There is a natural surjective complete homomorphism from the lattice £e(LI) of sub-e-varieties of LI to the lattice £(PS) of subvarieties of PS, the variety of all pseudosemilattices. Driven by the goal of understanding the structure of £e(LI), we study the lattice £(PS) in this dissertation. Understanding the structure of the free pseudosemilattices is a key step to studying £(PS). Thus, the first part of this dissertation deals with free pseudosemilattices. In particular, we give a solution to the word problem and construct some models for free pseudosemilattices. This solution to the word problem allows us to understand better the connections that exist amongst identities. A particular set of identities stands out, and we use it to define two infinite but countable sets of varieties of pseudosemilattices. The study of these varieties and of the connections amongst them gives us some insights on the structure of £(PS). We obtain some results especially about bases of identities, covers, and intersection of varieties in £(PS). Probably, the most surprising result states that every non-semigroup finite pseudosemilattice has infinite axiomatic rank, that is, any set of identities describing the equational theory of a non-semigroup finite pseudosemilattice contains identities with an arbitrarily large number of distinct variables.

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