On Finite Complete Presentations and Exact Decompositions of Semigroups

Abstract

We prove that given a finite (zero) exact right decomposition (M, T) of a semigroup S, if M is defined by a finite complete presentation, then S is also defined by a finite complete presentation. Exact right decompositions are natural generalizations to semigroups of coset decompositions in groups. As a consequence, we deduce that any Zappa–Szép extension of a monoid defined by a finite complete presentation, by a finite monoid, is also defined by such a presentation.

It is also proved that a semigroup M ⁰[A; I, J; P], where A and P satisfy some very general conditions, is also defined by a finite complete presentation.

Key Words:

• Complete rewriting system
• Complete presentation
• Factorizable semigroup
• Semidirect product
• Transversal
• Zappa–Szép product
• Zero exact decomposition
• Zero Rees matrix semigroup

2000 Mathematics Subject Classification:

• Primary 68Q42, 20M05
• Secondary 03D40, 20M50

ACKNOWLEDGEMENTS

We acknowledge with gratitude the suggestions made by both the referee and the editor, that strongly improved the article.

This work was developed within the project POCTI-ISFL-1-143 of CAUL, financed by FCT and FEDER. It is also part of the project “Semigroups and Languages – PTDC/MAT/69514/2006,” financed by FCT.

Notes

Communicated by M. Kambites.