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": \[ (\cap_{i \in w} Ω_{i}) \vee Θ_{t \odot s, s} = \cap_{k,r \in w} \big( (Ω_{k} \cap Ω_{r}) \vee Θ_{t \odot s, s} \big) \] for any family of congruences $\{{\Omega }_i:i\in w\}$, with $w$ a possibly infinite set.