Guram Bezhanishvili

In this paper, we introduce the notion of a Nachbin proximity on a bounded archimedean [Formula: see text]-algebra ([Formula: see text]-algebra). We prove that Gelfand duality lifts to yield a dual equivalence between the categories of uniformly complete [Formula: see text]-algebras equipped with a closed Nachbin proximity and of Nachbin spaces (compact ordered spaces). The key ingredients of the proof include appropriate generalizations of the Stone–Weierstrass theorem and Dieudonné’s lemma. We also develop an alternate approach by means of bounded archimedean [Formula: see text]-semialgebras ([Formula: see text]-algebras), from which we derive De Rudder–Hansoul duality.

There are numerous generalizations of the celebrated Priestley duality for bounded distributive lattices to the nondistributive setting. The resulting dualities rely on an earlier foundational work of such authors as Nachbin, Birkhoff–Frink, Bruns–Lakser, Hofmann–Mislove–Stralka, and others. We undertake a detailed comparative study of the existing dualities for arbitrary bounded (nondistributive) lattices, including supplying the dual description of bounded lattice homomorphisms where it was lacking. This is achieved by working with relations instead of functions. As a result, we arrive at a landscape of categories that provide various generalizations of the category of Priestley spaces. We provide explicit descriptions of the functors yielding equivalences of these categories, together with explicit dual equivalences with the category of bounded lattices and bounded lattice homomorphisms.

Esakia’s theorem states that Grzegorczyk’s logic is the greatest modal companion of intuitionistic propositional calculus. We prove that already the one-variable fragment of intuitionistic predicate calculus does not have a greatest modal companion, yielding that Esakia’s theorem fails in the monadic setting.

We introduce Esakia order-compactifications and study how they fit in the general theory of Priestley order-compactifications. We provide an analog of Dwinger's theorem by characterizing Esakia order-compactifications by means of special rings of upsets. These considerations naturally lead to the notion of a locally Esakia space, for which we prove that taking the largest Esakia order-compacification is functorial, thus obtaining an analog of Banaschewski's theorem.

Symbolic Logic, an international organization for furthering research in logic and the exchange of ideas among mathematicians, computer scientists, linguists, and others interested in this fi eld.

Priestley duality has diverse applications in various branches of mathematics. In this survey, we discuss its usefulness in pointfree topology. This is done by providing Priestley perspective on several key notions, including spatiality, sublocales, separation axioms, compactness, and local compactness. This approach yields a new perspective on a number of classic results in pointfree topology.

We investigate the set-theoretic strength of several maximality principles that play an important role in the study of modal and intuitionistic logics. We focus on well-known Fine’s and Esakia’s Maximality Principles, present two formulations of each, and show that the stronger formulations are equivalent to the Axiom of Choice (AC), while the weaker ones to the Boolean Prime Ideal Theorem (BPI).

Symbolic Logic, an international organization for furthering research in logic and the exchange of ideas among mathematicians, computer scientists, linguists, and others interested in this fi eld.