Distributive Lattices

We study various generalizations and weakenings of the Rasiowa-Sikorski Lemma for Boolean algebras. Building on previous work from Goldblatt, we extend the Rasiowa-Sikorski Lemma to co-Heyting algebras and modal algebras, and show how this yields completeness results for the corresponding non-classical first-order logics. Moreover, working without the full power of the Axiom of Choice, we generalize the framework of possibility semantics from Humberstone, and more recently Holliday, in order to provide choice-free representation theorems for distributive lattice, Heyting algebras and co-Heyting algebras. We also generalize a weaker version of the Rasiowa-Sikorski Lemma for Boolean algebras, known as Tarski's Lemma, to distributive lattices, HA's and co-HA's, and use these results to define a new semantics for first-order intuitionisitic logic.

In this paper, we solve an open problem in an special case. The problem is to give a characterization for Heyting algebras by means of fractions. Here, we give a representation for a class of Heyting algebras by means of fractions. Fractions on a bounded distributive lattice is a new algebraic structure, which was recently studied by the authors.