On a problem of M. Kambites regarding abundant semigroups

• Main Author: Araújo, João
• Publication Date: 2012
• Other Authors: Kinyon, Michael
• Format: Article
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://hdl.handle.net/10400.2/3817
• topic: Abundant semigroups; Adequate semigroups; Amiable semigroups; 20M10; 20M07; 20M20

A semigroup is regular if it contains at least one idempotent in each -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 -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 *-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 * 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 * and L*-class, must the idempotents commute? In this note, we provide a negative answer to this question.