Tree Algebras

Many important families of regular languages have effective characterizations in terms of syntactic monoids or syntactic semigroups (see e.g. [1]). Definition of the syntactic monoid (resp. syntactic semigroup) of a language L ⊆ X * (resp. a language L ⊆ X +) over an alphabet X requires regarding L as a subset of the free monoid X * (resp. free semigroup X +). On the other hand tree languages have traditionally been regarded as subsets of term algebras. So it appears natural to base the classification of regular tree languages over a ranked alphabet Σ on their syntactic algebras as was done by Steinby [4]. Also the syntactic monoids/semigroups of tree languages introduced by Thomas [5] have been used for characterizing families of tree languages. Ni-vat and Podelski's approach was, in a sense, a step further in this direction by treating (binary) trees as elements of an infinitely generated free monoid (see [2] and [3].) Here we consider another rather new formalism introduced by Wilke [6] in which trees are not directly viewed as elements of any algebraic structure but are represented by terms over a signature Γ which consists of six operation symbols involving three sorts Alphabet, Tree and Context. The trees thus represented are binary trees over a given label alphabet. A tree algebra is a Γ-algebra satisfying certain identities which identify (some) pairs of Γ-terms which represent the same tree. The syntactic tree algebra of tree language L is defined in a natural way. Its component of sort Tree is the syntactic