ORDERED AND J-TRIVIAL SEMIGROUPS AS DIVISORS OF SEMIGROUPS OF LANGUAGES

A semigroup of languages is a set of languages considered with (and closed under) the operation of catenation. In other words, semigroups of languages are subsemigroups of power-semigroups of free semigroups. We prove that a (finite) semigroup is positively ordered if and only if it is a homomorphic image, under an order-preserving homomorphism, of a (finite) semigroup of languages. Hence it follows that a finite semigroup is J-trivial if and only if it is a homomorphic image of a finite semigroup of languages.