General varieties of tree languages

Abstract

We consider varieties of tree languages which are not restricted to a fixed ranked alphabet, varieties of finite algebras that contain algebras of all finite types, and a matching notion of varieties of congruences. A variety theorem that yields isomorphisms between the lattices formed by these three types of varieties is proved. To achieve this, some basic universal algebra is suitably generalized and we define syntactic algebras so that two tree languages over any alphabets belong to the same varieties exactly in case their syntactic algebras are isomorphic. Many families of regular tree languages are shown to be varieties in our sense. In particular, we prove that every family of regular tree languages which can be characterized by syntactic monoids is such a variety, but that the converse does not hold.