Rectangular Algebras as Tree Recognizers

We consider finite rectangular algebras of finite type as tree recognizers.The type is represented by a ranked alphabet Σ. We determinethe varieties of finite rectangular Σ-algebras and show thatthey form a Boolean lattice in which the atoms are minimal varietiesof finite Σ-algebras consisting of projection algebras. We alsodescribe the corresponding varieties of Σ-tree languages andcompare them with some other varieties studied in the literature.Moreover, we establish the solidity properties of these varietiesof finite algebras and tree languages. Rectangular algebras havebeen previously studied by R. Pöschel and M. Reichel 1993, andwe make use of some of their results.