Semigroups with Idempotent stabilizers and Applications to Automata Theory

We show that every finite semigroup is a quotient of a finite semigroup in which every right stabilizer satisfies the identities x = x and xy = xyx. This result has several consequences. We first use it together with a result of I. Simon on congruences on paths to obtain a purely algebraic proof of a deep theorem of McNaughton on infinite words. Next, we give an algebraic proof of a theorem of Brown on a finiteness condition for semigroups.