Theorem 5.84.
Let \(K, L \in \Cong(T)\) be congruences. Then their product \(KL\) is also a congruence.
Proof
By (5) of Proposition 5.35, it suffices to show that it is closed under diagonals, i.e. \(KL \subseteq D(KL)\). Recall that any acyclic class \(M\) decomposes as \(M = M^{\mono} \vee M^{\epi}\). Applying this to \(K\) and \(L\), we can distribute the product:
\[KL = (K^{\epi} \vee K^{\mono})(L^{\epi} \vee L^{\mono}) = K^{\epi}L^{\epi} \vee K^{\epi}L^{\mono} \vee K^{\mono}L^{\epi} \vee K^{\mono}L^{\mono}.\]
The product distributes over joins by Proposition 5.75. Using that \(K^{\epi} = \EffEpi \cdot K\) and \(L^{\epi} = \EffEpi \cdot L\), we see that the first three terms are contained in \(\EffEpi \cdot K \cdot L\), which by Corollary 5.83 is contained in \(D(KL)\). It thus remains to show that \(K^{\mono} L^{\mono} \subseteq D(KL)\). The product \(K^{\mono}L^{\mono}\) is monogenic by Corollary 5.76, and it is contained in \(KL\). Hence it is contained in the largest monogenic acyclic class below \(KL\), namely \((KL)^{\mono}\). By Lemma 5.60, the latter is a congruence and therefore closed under diagonals. Thus \((KL)^{\mono} \subseteq D(KL)\), finishing the proof.