The Kernel Relation for a Completely Regular Semigroup

✍ Scribed by M. Petrich

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 872 KB
• Volume: 172
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1995.1050

✦ Synopsis

The kernel relation for a regular semigroup (S) identifies two congruences on (S) if they have the same kernel. It is always a complete (\wedge)-congruence on the congruence lattice (\mathscr{C}(S)) of (S). We give a great number of equivalent conditions on a completely regular semigroup (S), one of which is that (K) be a (complete) congruence on (\mathscr{C}(S)). These conditions bear upon minimal congruences identifying two comparable elements of (S), variants of (\theta)-modularity, the mappings (\rho \rightarrow \operatorname{ker} \rho), (\rho \rightarrow \rho_{K}, \rho \rightarrow \rho \cap \mathscr{H}) being (complete) (\vee)-homomorphisms, least group congruences on certain completely simple semigroups, certain subgroups of (S), and the standard representation of (S). The paper concludes with a discussion of special cases. 1995 Academic Press, Inc.