ABSTRACT
So(a) = yo(b) implies [49, [b] 93 and thus (a, b) E 03 for all j E J. Therefore (a, b) c n 0, = AA and so a = b. This proves the isomorphism of A
JEJ and (p(A). Moreover, if pk : H (A1613) —> AA denotes the k-th projection jEJ mapping, then by the definition of (p we have pk(cp(A)) = AlOk for all k E J. Therefore co(A) is a subdirect product of the algebras A193. •
We remark that the converse of this theorem is also true. If A is isomorphic to a subdirect product of a family (Aj)j E J of algebras, then there exists a family of congruence relations on A whose intersection is the relation AA. We leave the proof as an exercise for the reader.
