An equivalent of subdirect irreducibility for semigroups

Let S be a semigroup. Prove that S is a subdirect product of the family of semigroups \{S_a\}_A indexed by A if and only if for every a \in A, there is a surjective semigroup homomorphism \varphi_a : S \rightarrow S_a such that the family of induced congruences (that is, the kernels of the \varphi_a) separate the elements of S.


First, suppose \psi : S \rightarrow \prod_A S_a is a subdirect product; then by definition the composites \varphi_a = \pi_a \circ \psi : S \rightarrow S_a are surjective semigroup homomorphisms. The family of kernels of the \varphi_a separate the elements of S by Proposition II.1.4.

Conversely, if we have such a family of homomorphisms, then by Proposition II.1.4, S is a subdirect product of the family \{S/\mathsf{ker}\ \varphi_a\}_A. By the First Isomorphism Theorem for semigroups (proved here), S/\mathsf{ker}\ \varphi_a \cong S_a, so that S is a subdirect product of the S_a.

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