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Abstract
When a semigroup has a unary operation, it is possible to define two binary operations, namely, left and right division. In addition, it is well known that groups can be defined in terms of those two divisions. The aim of this article is to extend those results to other classes of unary semigroups. In the first part of the article, we provide characterizations for several classes of unary semigroups, including (a special class of) E-inversive, regular, completely regular, inverse, Clifford, etc., in terms of left and right division. In the second part, we solve a problem that was posed elsewhere. The article closes with a list of open problems.
ACKNOWLEDGMENTS
We are pleased to acknowledge the assistance of the automated deduction tool \textscProver9 and the finite model builder \textscMace4, both developed by McCune [Citation8]. The first author was partially supported by FCT and FEDER, Project POCTI-ISFL-1-143 of Centro de Algebra da Universidade de Lisboa, by FCT and PIDDAC through the project PTDC/MAT/69514/2006, by PTDC/MAT/69514/2006 Semigroups and Languages, and by PTDC/MAT/101993/2008 Computations in groups and semigroups.
Notes
Communicated by V. Gould.