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Abstract
A semigroup is regular if it contains at least one idempotent in each ℛ-class and in each ℒ-class. A regular semigroup is inverse if it satisfies either of the following equivalent conditions: (i) there is a unique idempotent in each ℛ-class and in each ℒ-class, or (ii) the idempotents commute. Analogously, a semigroup is abundant if it contains at least one idempotent in each ℛ*-class and in each ℒ*-class. An abundant semigroup is adequate if its idempotents commute. In adequate semigroups, there is a unique idempotent in each ℛ* and ℒ*-class. M. Kambites raised the question of the converse: in a finite abundant semigroup such that there is a unique idempotent in each ℛ* and ℒ*-class, must the idempotents commute? In this note, we provide a negative answer to this question.
ACKNOWLEDGMENT
We are pleased to acknowledge the assistance of the finite model builder Mace4 and the automated deduction tool Prover9, both developed by McCune [Citation6]. We initially found small examples using Mace4 and then reverse engineered them into the examples presented here. Prover9 was helpful in working out the proofs in Section 2.
The first author was partially supported by FCT and FEDER, Project POCTI-ISFL-1-143 of Centro de Algebra da Universidade de Lisboa, and by FCT and PIDDAC through the project PTDC/MAT/69514/2006.
Notes
Communicated by V. Gould.