Narrow Results

Straubing deduced from Simon's theorem that monoids of reflexive relations as well as monoids of order preserving extensive transformations generate the pseudovariety of -trivial monoids. We refine these results by showing that each pseudovariety in Simon's hierarchy of -trivial monoids is generated by a single relation/transformation monoid. From this and from some results by Blanchet–Sadri we obtain a complete solution of the finite basis problem for Straubing's monoids.

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 -trivial if and only if it is a homomorphic image of a finite semigroup of languages.