Christoph Wolf:
Many-Sorted Algebras in Congruence Modular Varieties

(Algebra Universalis 36 (1996), 66-80)


We present several basic results on many-sorted algebras, most of them only valid in congruence modular varieties. We describe a connection between the properties of many-sorted varieties and those of varieties of one sort and give some results on functional completeness, the commutator and Abelian algebras.


Many-sorted algebras were introduced by P.J. Higgins as an extension of one-sorted algebras. He examined properties of free algebras and varieties. Since then the theory of many-sorted algebras has been influenced mainly by the requirements of computer science. There definitions like abstract data types and algebraic specifications were placed in the foreground. In contrast to that, this paper concentrates on results which are obtained by methods and motivated by goals which are efficient in the structure theory of one-sorted algebras. For this congruence modular varieties seem to be a good starting point.

First we give some basic definitions and results. In the second section we define for a many-sorted algebra the restriction to one sort which plays an important role in the rest of this paper. We show that those properties of many-sorted varieties given by congruence equations can be characterized by those of varieties generated by restriction to the sorts. Then we give conditions for many-sorted algebras to be functionally complete. In the last section we compare the many-sorted to the one-sorted commutator and characterize many-sorted Abelian algebras in congruence modular varieties.

We assume the reader to be familiar with the theory of one-sorted universal algebra.

Back .