(Algebra Universalis 36 (1996), 66-80)
Many-Sorted Algebras in Congruence Modular Varieties
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
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
We assume the reader to be familiar with the theory of one-sorted