The pairwise distributive law of semilattice congruences
Fernando Martín-Maroto, Antonio Ricciardo, Gonzalo G. de Polavieja · Algebra Universalis · 2026
We show that the congruence lattice of a semilattice satisfies a form of distributivity relative to principal congruences of the form $$ \Theta _{t \odot s, s}.$$ Particularly, we establish that semilattice congruences obey the “pairwise distributive law”: $$ (\cap _{i \in w} \Omega _{i}) \vee \Theta _{t \odot s, s} = \cap _{k,r \in w} \big ( (\Omega _{k} \cap \Omega _{r}) \vee \Theta _{t \odot s, s} \big ) $$ for any two elements t and s and any family of congruences $$\{ \Omega _{i}: i\in w \},$$ with w a possibly infinite set.