Rectangular algebras as tree recognizers

Abstract

We consider finite rectangular algebras of finite type as tree recognizers. The type is represented by a ranked alphabet Σ. We determine the varieties of finite rectangular Σ-algebras and show that they form a Boolean lattice in which the atoms are minimal varieties of finite Σ-algebras consisting of projection algebras. We also describe the corresponding varieties of Σ-tree languages and compare them with some other varieties studied in the literature. Moreover, we establish the solidity properties of these varieties of finite algebras and tree languages. Rectangular algebras have been previously studied by R. Pöschel and M. Reichel (1993), and we make use of some of their results.