The pairwise distributive law of semilattice congruences

Abstract

We show that the congruence lattice of a semilattice satsifies 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 family of congruences $\{ \Omega_{i} : i\in w \}$, with $w$ a possibly infinite set.