The Congruence Extension Property and Related Topics in Semigroups.

Abstract

A semigroup has the congruence extension property (CEP) provided that each congruence on each subsemigroup can be extended to a congruence on the semi-group. This property, the ideal extension property (IEP), and other related concepts are studied from both an algebraic and a topological perspective in this work. A characterization of semigroups with CEP is given in terms of the lattice of congruences. A similar result is obtained for IEP. Semigroups in which the relation "is an ideal of" is transitive (t-semigroups) are explored. It is shown that each of CEP and IEP implies this condition and that these are all equivalent for cyclic semigroups. Semigroups in which every subsemigroup is an ideal of some ideal (m-semigroups) are discussed. It is obtained that m-semigroups are periodic semigroups with zero and index less than or equal to 5. Those commutative m-semigroups with index less than or equal to 3 are characterized. The majority of these results are topologized for compact semigroups. Compact completely simple semigroups with CEP are characterized. A construction is given which yields an alternative proof for the known result in the algebraic case and is amenable to direct extension to the topological result. Additionally, subsemigroups of a completely simple semigroup with torsion subgroups are characterized. It is shown that CEP is retained by continuous homomorphic images of compact completely simple semigroups with CEP.