Let be a semigroup. Prove that is a subdirect product of the family of semigroups indexed by if and only if for every , there is a surjective semigroup homomorphism such that the family of induced congruences (that is, the kernels of the ) separate the elements of .
First, suppose is a subdirect product; then by definition the composites are surjective semigroup homomorphisms. The family of kernels of the separate the elements of by Proposition II.1.4.
Conversely, if we have such a family of homomorphisms, then by Proposition II.1.4, is a subdirect product of the family . By the First Isomorphism Theorem for semigroups (proved here), , so that is a subdirect product of the .