Differents types of Congruences in Direct Products

In order to generalize the notion of a product of congruences in a direct product of similar algebras in the case of infinitely many factors, we introduce the concept of decomposable congruence: a congruence ϑ of U = Πi ϵI Ui is said to be decomposable if there exist a family (ϑi)iϵi of congruences of the algebras i and a filter G on I such that xϑy iff . We first consider products of Boolean algebras, defining in a similar manner decomposable filters. After giving sufficient conditions for a filter to be decomposable (Theorems 1 and 2), we exhibit (Example 1) a non-decomposable filter of a product ΠiϵIBi such the set { is bounded by a natural number (under this hypothesis, on the contrary, any ultrafilter is decomposable). Two topological characterizations of decomposable filters (Theorems 4 and 5) are also given. In Section 4 we compare decomposable congruences with filtral and ideal ones and discuss the problem of defining a suitable filter G on I starting from a congruence ϑ of (this is always possible for Boolean algebras). In Section 5 we examine some properties of decomposable congruences (permutability, distributivity,…), generalizing some results relative to filtral congruences. Finally, we give a classification of filtral, decomposable and ideal congruences, showing in particular that decomposable classes are just filtral ones.