Equivalent characterizations of subdirect irreducibility

Let S be a nontrivial semigroup. Show that the following are equivalent.

  1. S is subdirectly irreducible.
  2. The set of proper congruences on S is closed under arbitrary intersections.
  3. S has a \subseteq-least proper congruence.

We will follow the strategy (1) \Rightarrow (2) \Rightarrow (3) \Rightarrow (1). Note that the condition that S is nontrivial is required to have the set \mathsf{Q}(S) of nonidentity congruences be nonempty.

(1) \Rightarrow (2): Suppose S is a subdirectly irreducible semigroup. Let Q \subseteq \mathsf{Q}(S) be a nonempty set of proper congruences on S, and consider the product \prod_{\sigma \in Q} S/\sigma.

Suppose \bigcap Q = \Delta. (That is, suppose the intersection over Q is the identity congruence.) By II.1.3, the family of congruences in Q separate S. By II.1.4, S is a subdirect product of \{S/\sigma\}_Q via the map s \mapsto ([s]_\sigma). Since S is subdirectly irreducible, then for some congruence \sigma, the map s \mapsto [s]_\sigma is an isomorphism. Clearly then \sigma = \Delta, a contradiction. So \bigcap Q \supsetneq \Delta.

(2) \Rightarrow (3): If \mathsf{Q}(S) is the set of all proper (i.e. nontrivial) congruences on S, then (by our hypothesis) \bigcap \mathsf{Q}(S) = \tau \in \mathsf{Q}(S) is a nontrivial congruence. Certainly if \sigma \in \mathsf{Q}(S) then \tau \subseteq \sigma, so that \tau is \subseteq-least among the elements of \mathsf{Q}(S).

(3) \Rightarrow (1): Suppose S has a \subseteq-least proper congruence \tau. Suppose further that \varphi : S \rightarrow \prod_B S_\beta is a subdirect product; that is, \varphi is injective and each \pi_\beta \circ \varphi is surjective. By II.1.4, the family of congruences \mathsf{ker}\ \pi_\beta \circ \varphi separate the elements of S. By II.1.3, we have \bigcap \mathsf{ker}\ \pi_\beta \circ \varphi = \Delta. Since S has a \subseteq-least proper congruence, one of the \mathsf{ker}\ \pi_\beta \circ \varphi must be \Delta. So \pi_\beta \circ \varphi : S \rightarrow S_\beta is an isomorphism.

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