Publication Date

12-2-1969

Abstract

In his paper, "A Semigroup Approach to Lattices, 11 (Canad . Jour. Math., Vol. 18 (1966)), M. F. Janowitz showed that every lattice with 0 and 1 is isomorphic to the lattice formed by the left annihilators of a suitably chosen Baer semigroup when they are ordered under set inclusion. A Baer semigroup, s~ was then said to coordinatize a lattice, L, if Lis isomorphic to the lattice of left annihilators of S. This dissertation presents a similar semigroup approach to lattices by showing that a poset, P, with O and 1 forms a meet semilattice if and only if Pis isomorphic to (can be coordinatized by) the meet semilattice of right annihilators formed by a class of semigroups with annihilator structure only slightly weaker than that of a Baer semigroup. These are called right Baer semigroups.

The axioms for a right Baer semigroup are developed from the more general notion of a right annihilator semigroup, which is a semigroup with O in which the right annihilator of every element is the right principal ideal generated by an idempotent, and the left annihilator of every idempotent which generates a right annihilator is the left principal ideal generated by an idempotent. It is then shown that the set of right annihilators of a right Baer semigroup form a meet semilattice with O and 1, and that every meet semilattice with O and 1, P, is isomorphic to the lattice of right annihilators of the right Baer semigroup generated by all the 1-preserving meet endomorphisms on P.

Finally, necessary and sufficient conditions for the right annihilators of a right Baer semigroup to form a lattice are generated and the classes of right Baer semigroups which coordinatize modular, distributive, and implicative lattices and Boolean algebras are characterized. The characterizations of Baer semigroups which coordinatize the former three types of lattices have not yet been produced.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Melvin J.

Second Committee Member

Julius Rubin Blum

Third Committee Member

Lambert Herman Koopmans

Language

English

Document Type

Dissertation

Download

Share

COinS