Transitivity of subdirect products

Let S be a semigroup. Suppose S is a subdirect product of the family \{S_a\}_A, and that for each a \in A, S_a is a subdirect product of the family \{S_{b,a}\}_{B_a}. Show that S is a subdirect product of the family \{S_{b,a}\}_{B_a,A}. (Roughly speaking, subdirect productness is transitive.)


By definition, we have injective semigroup homomorphisms \varphi : S \rightarrow \prod_A S_a and \psi_a : S_a \rightarrow \prod_{B_a} S_{b,a} which are surjective in each component. Now for each a \in A and b \in B_a, \zeta_{b,a} = \pi_{b,a} \circ \psi_a \circ \pi_a \circ \varphi : S \rightarrow S_{b,a} is a surjective homomorphism (being the composite of two surjective homomorphisms).

Consider the induced family of congruences \sigma_{b,a} = \mathsf{ker}\ \zeta_{b,a}. Let x,y \in S; since S is a subdirect product of the S_a, there exist a_1, a_2 such that (\pi_{a_1} \circ \varphi)(x) \neq (\pi_{a_2} \circ \varphi)(y), and so in particular x and y are separated by the congruences \sigma_{b,a}. By Proposition II.1.4, S is a subdirect product of the family S/\sigma_{b,a}, which are naturally isomorphic to S_{b,a} by the First Isomorphism Theorem. So S is a subdirect product of the S_{b,a}.

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