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In this talk, we will explore the concept of i-Fitting-Formation of Clifford semigroups.
A Clifford semigroup is an inverse semigroup that is a union of groups. A class of Clifford semigroups is an i-formation if it is closed for idempotent separating quotients and for finitary subdirect products with idempotent separating projections. Dually, a Fitting class is a class of Clifford semigroups that is closed for normal subsemigroups and for Clifford semigroups generated by two normal subsemigroups in the class. Motivated by [3] and [1], in [2], using the concepts of i-formation and i-Fitting class, we established bijections between classes of inverse semigroups (and in particular of Clifford semigroups), of congruences and of languages. This will be the starting point of this talk.
As in group theory, after looking at formations and Fitting classes separately, a natural next step is to consider i-Fitting-formations of Clifford semigroups, i.e., classes that are simultaneously i- formation and Fitting class. In this talk, we will present the corresponding concepts for congruences and for languages, establishing bijections between the respective classes and with i-Fitting-formations of Clifford semigroups.
This is joint work with Gracinda Gomes.
[1] A. Ballester-Bolinches, E. Cosme Llópez, R. Esteban-Romero, and J. Rutten. Formations of monoids, congruences, and formal languages. Scientific Annals of Computer Science, 25:171-209, 2015.
[2] G. M. S. Gomes and A.-C. C. Monteiro. Formations and i-fitting classes of inverse semigroups, congruences and languages. to appear in Semigroup Forum.
[3] D. Therien. Classification of Regular Languages by Congruences. PhD thesis, 1980. AAI0533628.
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