  We study varieties that contain unranked tree languages over all alphabets.
Trees are labeled with symbols from two alphabets, an unranked operator
alphabet and an alphabet used for leaves only. Syntactic algebras of unranked
tree languages are defined similarly as for ranked tree languages, and an
unranked tree language is shown to be recognizable iff its syntactic algebra is
regular, i.e., a finite unranked algebra in which the operations are defined by
regular languages over its set of elements. We establish a bijective
correspondence between varieties of unranked tree languages and varieties of
regular algebras. For this, we develop a basic theory of unranked algebras in
which algebras over all operator alphabets are considered together. Finally, we
show that the natural unranked counterparts of several general varieties of
ranked tree languages form varieties in our sense.
  This work parallels closely the theory of general varieties of ranked tree
languages and general varieties of finite algebras, but many nontrivial
modifications are required. For example, principal varieties as the basic
building blocks of varieties of tree languages have to be replaced by what we
call quasi-principal varieties, and we device a general scheme for defining
these by certain systems of congruences.
