On Pseudovarieties of Forest Algebras

Forest algebras are defined for investigating languages of forests [ordered sequences] of unranked trees, where a node may have more than two [ordered] successors. They consist of two monoids, the horizontal and the vertical, with an action of the vertical monoid on the horizontal monoid, and a complementary axiom of faithfulness. In the study of forest algebras one of the main difficulties is how to handle the faithfulness property. A pseudovariety is a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. We tried to adapt in this context some of the results in the theory of semigroups, specially the studies on relatively free profinite semigroups, which are an important tool in the theory of pseudovarieties of semigroups. We define a new version of syntactic congruence of a subset of the free forest algebra, not just a forest language. This new version is the natural extension of the syntactic congruence for monoids in the case of forest algebras and is used in the proof of an analog of Hunter’s Lemma. We show that under a certain assumption the two versions of syntactic congruences coincide. We adapt some results of Almeida on metric semigroups to the context of forest algebras. We show that the analog of Hunter’s Lemma holds for metric forest algebras, which leads to the result that zero-dimensional compact metric forest algebras are residually finite. We show an analog of Reiterman’s Theorem, which is based on a study of the structure profinite forest algebras.