The pairwise distributive law of semilattice congruences
Fernando Martin-Maroto, Antonio Ricciardo, Gonzalo G. de Polavieja
TL;DR
This work analyzes the congruence lattice of semilattices, showing that although semilattice congruences do not satisfy pure lattice identities, they obey a pairwise distributive law relative to principal congruences $Θ_{t \odot s, s}$. The authors prove the law first for maximal congruences and then extend to arbitrary congruences by expressing them as intersections of maximal congruences, aided by a detailed analysis of $Θ_{t \odot s, s}$. The results include a general identity $(\cap_{i\in w} \Omega_i) \vee Θ_{t \odot s, s} = \bigcap_{k,r\in w} ((\Omega_k \cap \Omega_r) \vee Θ_{t \odot s, s})$ and a characterization of principal congruences, along with independent proofs and discussions of the role of maximal congruences and the crossing technique. The findings deepen understanding of semilattice congruence lattices and provide tools for analyzing their distributive-like behavior in conjunction with principal congruences.
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.