A Stone duality theorem for arbitrary lattices
Andrés Ríos · arXiv (Cornell University) · 2023
We establish a Stone duality theorem for arbitrary lattices, without assuming distributivity or boundedness. Rather than extending Priestley-style ordered representations, our construction follows the modern Stone-Balbes-Dwinger viewpoint: a lattice is recovered from distinguished open subsets of a spectrum. Since prime ideals are no longer sufficient in the non-distributive setting, we replace them by comaximal pairs and associate a bitopological spectrum to each lattice. The two topologies arise from two natural Stone-like maps: one encodes the join-semilattice structure, while the other encodes the meet-semilattice structure. These two maps coincide precisely in the distributive case, showing that the single topology of the classical Stone-Balbes-Dwinger representation is the collapsed form of a genuinely bitopological construction. We introduce pairwise Balbes-Dwinger spaces as the corresponding class of bitopological spaces and prove that each lattice is naturally isomorphic to the lattice of essential subsets of its bitopological spectrum. This yields a representation theorem for arbitrary lattices. We then study functoriality and prove a dual equivalence between lattices with quasi-proper homomorphisms and pairwise Balbes-Dwinger spaces with the appropriate morphisms. In the distributive case, the two topologies coincide and the classical Stone-Balbes-Dwinger duality is recovered.