On Definite and Nilpotent DR Tree Languages

The sets recognized by deterministic root-to-frontier tree recognizers, the DR tree languages, are determined by their path languages. A path language of a tree language T consists of the words describing the paths leading from the root of a tree in T to its leaves labelled with a given leaf symbol. The greatest congruence saturating the path languages yields the syntactic path monoid of T which is finite for a path closed T exactly in case T is a DR tree language. We introduce the concepts of definite and nilpotent DR tree languages and, by means of syntactic path semigroups, characterize them.