Semigroups whith idempotent stabilizers and applications to automata theory

Bertrand Le Saec (1) , Jean-Eric Pin (2) , Pascal Weil (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

We show that every finite semigroup is a quotient of a finite semigroup in which every right stabilizer satisfies the identities x = x^2 and xy = xyx. This result has several consequences. We first give a geometrical application : every finite transformation semigroup has a fixpoint-free covering (a transformation semigroup is fixpoint-free if every element which stabilizes a point is idempotent). Next we use our result and a result of I. Simon on congruences on paths to obtain a purely algebraic proof of a deep theorem of McNaughton on infinite words. Finally, we give an algebraic proof of a theorem of Brown on a finiteness condition for semigroups.

Mots clés: finite semigroup; infinite words; stabilizer

Domaines: Autre [cs.OH]

Bertrand Le Saec, Jean-Eric Pin, Pascal Weil. Semigroups whith idempotent stabilizers and applications to automata theory. International Journal of Algebra and Computation, 1991, 1, pp.291-314. ⟨hal-00019824⟩