The pairwise distributive law of semilattice congruences

2025/11/02 by Fernando Martín-Maroto, Martin-Maroto, Fernando, Antonio Ricciardo +3
Computer Science · Decision Sciences · #Advanced Algebra and Logic #Commutative Algebra (math.AC) #FOS: Mathematics #Multi-Criteria Decision Making #Rings and Algebras (math.RA) #Rough Sets and Fuzzy Logic

paper · pdf · doi:10.48550/arxiv.2511.00892

openalex publication_date 2025/11/02 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28

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": (∩_{i ∈ w} Ω_i) \vee Θ_{t \odot s, s} = ∩_{k,r ∈ w} ( (Ω_k ∩ Ω_r) \vee Θ_{t \odot s, s} ) for any family of congruences \ Ω_i : i∈ w \, with w a possibly infinite set.

Citations

• Finite Atomized Semilattices

Related

• Revisiting the representation theorem of finite distributive lattices with principal congruences
• A short proof of the congruence representation theorem for semimodular lattices
• Notes on congruence lattices and lamps of slim semimodular lattices
• An extension theorem for planar semimodular lattices
• Representing finite distributive lattices as congruence lattices of lattices