The pairwise distributive law of semilattice congruences

Fernando Martin-Maroto, Antonio Ricciardo, Gonzalo G. de Polavieja

Abstract: We show that the congruence lattice of a semilattice satsifies a form of distributivity relative to principal congruences of the form $ Θ_{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 $\{ Ω_{i} : i\in w \}$, with $w$ a possibly infinite set.

External IDs:doi:10.48550/arxiv.2511.00892