## LSU Historical Dissertations and Theses

1993

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

John A. Hildebrant

#### Abstract

A semigroup has the congruence extension property (CEP) provided that each congruence on each subsemigroup of S extends to a congruence on S. The ideal extension property (IEP) for semigroups is defined analogously. A characterization of commutative semigroups with IEP is given in terms of multiplicative conditions within and between the archimedean components of the semigroup. A similar characterization of commutative semigroups with CEP is sought. Toward this end, archimedean semigroups with CEP are characterized in terms of multiplicative structure and a number of necessary conditions on multiplication between the archimedean components of a commutative semigroup with CEP are established. In the topological setting, compact commutative unipotent semigroups with IEP and CEP respectively are characterized. Relative to the question of whether CEP is preserved by homomorphisms, necessary and sufficient conditions for a given (continuous) homomorphic image to retain CEP are given. It is proved that the homomorphic image of an archimedean semigroup with CEP has CEP and that Rees quotients of compact semigroups with CEP have CEP. An ideal semigroup is a semigroup in which each congruence is determined by an ideal. A characterization of commutative ideal semigroups is given which provides a description of the ${\cal H}$-order graph (or divisibility ordering) of such a semigroup. Consequently, the ${\cal H}$-order graphs of commutative ideal semigroups with IEP and CEP respectively are completely described. Commutative semigroups with CEP whose congruences form a chain are characterized as an unexpected corollary.

168

COinS