arXiv subjects
Publications and source records attributed to Luca Carai.
We connect Priestley duality for distributive lattices and its generalization to distributive meet-semilattices to Hofmann-Mislove-Stralka duality for semilattices. Among other things, this involves consideration of various morphisms between algebraic frames. We also show how Stone duality for boolean algebras and generalized boolean algebras fits as a particular case of the general picture we develop.
$\mathsf{S5}$-subordination algebras are a natural generalization of de Vries algebras. Recently it was proved that the category $\mathsf{SubS5^S}$ of $\mathsf{S5}$-subordination algebras and compatible subordination relations between them is equivalent to the category of compact Hausdorff spaces and closed relations. We generalize MacNeille completions of boolean algebras to the setting of $\mathsf{S5}$-subordination algebras, and utilize the relational nature of the morphisms in $\mathsf{SubS5^S}$ to prove that the MacNeille completion functor establishes an equivalence between $\mathsf{SubS5^S}$ and its full subcategory consisting of de Vries algebras. We also show that the functor that associates to each $\mathsf{S5}$-subordination algebra the frame of its round ideals establishes a dual equivalence between $\mathsf{SubS5^S}$ and the category of compact regular frames and preframe homomorphisms. Our results are choice-free and provide further insight into Stone-like dualities for compact Hausdorff spaces with various morphisms between them. In particular, we show how they restrict to the wide subcategories of $\mathsf{SubS5^S}$ corresponding to continuous relations and continuous functions between compact Hausdorff spaces.
Combining tools from category theory, model theory, and non-standard analysis we extend Baker-Beynon dualities to the classes of all Abelian $\ell$-groups and all Riesz spaces (also known as vector lattices). The extended dualities have a strong geometrical flavor, as they involve a non-standard version of the category of polyhedral cones and piecewise (homogeneous) linear maps between them. We further show that our dualities are induced by the functor $\mathtt{Spec}$, once it is understood how to endow it with "coordinates" in some ultrapower $\mathcal{U}$ of $\mathbb{R}$. This also allows us to characterize the topological spaces arising as spectra of Abelian $\ell$-groups and Riesz spaces as certain subspaces of $\mathcal{U}^κ$ endowed with the Zariski topology given by definable functions in their respective languages. Furthermore, we provide some applications of the extended duality by characterizing, in geometrical terms, semisimplicity, Archimedeanity, and the existence of weak and strong order-units. Finally, we show that our dualities afford a neat and simpler proof of Panti's celebrated characterization of the prime ideals in free Abelian $\ell$-groups and Riesz spaces.
We generalize the classic Vietoris endofunctor to the category of compact Hausdorff spaces and closed relations. The lift of a closed relation is done by generalizing the construction of the Egli-Milner order. We describe the dual endofunctor on the category of de Vries algebras and subordinations. This is done in several steps, by first generalizing the construction of Venema and Vosmaer to the category of boolean algebras and subordinations, then lifting it up to $\mathsf{S5}$-subordination algebras, and finally using MacNeille completions to further lift it to de Vries algebras. Among other things, this yields a generalization of Johnstone's pointfree construction of the Vietoris endofunctor to the category of compact regular frames and preframe homomorphisms.
We give an alternative, more geometric, proof of the well-known Joyal-Tierney Theorem in locale theory by utilizing Priestley duality for frames.
We introduce the category of Heyting frames and show that it is equivalent to the category of Heyting algebras and dually equivalent to the category of Esakia spaces. This provides a frame-theoretic perspective on Esakia duality for Heyting algebras. We also generalize these results to the setting of Brouwerian algebras and Brouwerian semilattices by introducing the corresponding categories of Brouwerian frames and extending the above equivalences and dual equivalences. This provides a frame-theoretic perspective on generalized Esakia duality for Brouwerian algebras and Brouwerian semilattices.
We develop a unified approach to Gelfand and de Vries dualities for compact Hausdorff spaces, which is based on appropriate modifications of the classic results of Dieudonné (analysis), Dilworth (lattice theory), and Kat{ě}tov-Tong (topology).
Let CABA be the category of complete and atomic boolean algebras and complete boolean homomorphisms, and let CSL be the category of complete meet-semilattices and complete meet-homomorphisms. We show that the forgetful functor from CABA to CSL has a left adjoint. This allows us to describe an endofunctor H on CABA such that the category Alg(H) of algebras for H is dually equivalent to the category Coalg(P) of coalgebras for the powerset endofunctor P on Set. As a consequence, we derive Thomason duality from Tarski duality, thus paralleling how Jónsson-Tarski duality is derived from Stone duality.
We devise three strategies for recognizing admissibility of non-standard inference rules via interpolation, uniform interpolation, and model completions. We apply our machinery to the case of symmetric implication calculus $\mathsf{S^2IC}$, where we also supply a finite axiomatization of the model completion of its algebraic counterpart, via the equivalent theory of contact algebras. Using this result we obtain a finite basis for admissible $Π_2$-rules.
By Priestley duality, each bounded distributive lattice is represented as the lattice of clopen upsets of a Priestley space, and by Esakia duality, each Heyting algebra is represented as the lattice of clopen upsets of an Esakia space. Esakia spaces are those Priestley spaces that satisfy the additional condition that the downset of each clopen is clopen. We show that in the metrizable case Esakia spaces can be singled out by forbidding three simple configurations. Since metrizability yields that the corresponding lattice of clopen upsets is countable, this provides a characterization of countable Heyting algebras. We show that this characterization no longer holds in the uncountable case. Our results have analogues for co-Heyting algebras and bi-Heyting algebras, and they easily generalize to the setting of p-algebras.
We show that intuitionistic quantifiers admit the following temporal interpretation: $\forall x A$ is true at a world $w$ iff $A$ is true at every object in the domain of every future world, and $\exists x A$ is true at $w$ iff $A$ is true at some object in the domain of some past world. For this purpose we work with a predicate version of the well-known tense propositional logic $\sf S4.t$. The predicate logic $\sf Q^\circ S4.t$ is obtained by weakening the axioms of the standard predicate extension $\sf QS4.t$ of $\sf S4.t$ along the lines Corsi weakened $\sf QK$ to $\sf Q^\circ K$. The Gödel translation embeds the predicate intuitionistic logic $\sf IQC$ into $\sf QS4$ fully and faithfully. We provide a temporal version of the Gödel translation and prove that it embeds $\sf IQC$ into $\sf Q^\circ S4.t$ fully and faithfully; that is, we show that a sentence is provable in $\sf IQC$ iff its translation is provable in $\sf Q^\circ S4.t$. Faithfulness is proved using syntactic methods, while we prove fullness utilizing the generalized Kripke semantics of Corsi.
In a recent paper we showed that intuitionistic quantifiers admit the following temporal interpretation: "always in the future" (for $\forall$) and "sometime in the past" (for $\exists$). In this paper we study this interpretation for the monadic fragment $\sf MIPC$ of the intuitionistic predicate logic. It is well known that $\sf MIPC$ is translated fully and faithfully into the monadic fragment $\sf MS4$ of the predicate $\sf S4$ (Gödel translation). We introduce a new tense extension of $\sf S4$, denoted by $\sf TS4$, and provide an alternative full and faithful translation of $\sf MIPC$ into $\sf TS4$, which yields the temporal interpretation of monadic intuitionistic quantifiers mentioned above. We compare this new translation with the Gödel translation by showing that both $\sf MS4$ and $\sf TS4$ can be translated fully and faithfully into a tense extension of $\sf MS4$, which we denote by $\sf MS4.t$. This is done by utilizing the algebraic and relational semantics for the new logics introduced. As a byproduct, we prove the finite model property (fmp) for $\sf MS4.t$ and show that the fmp for the other logics involved can be derived as a consequence of the fullness and faithfulness of the translations considered.
It is a classic result in modal logic that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality. Our goal is to further generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space. Our starting point is the well-known Gelfand duality between the category $KHaus$ of compact Hausdorff spaces and the category $ubal$ of uniformly complete bounded archimedean $\ell$-algebras. We endow a bounded archimedean $\ell$-algebra with a modal operator, which results in the category $mbal$ of modal bounded archimedean $\ell$-algebras. Our main result establishes a dual adjunction between $mbal$ and the category $KHK$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between $KHK$ and the reflective subcategory $mubal$ of $mbal$ consisting of uniformly complete objects of $mbal$. This generalizes both Gelfand duality and the duality for modal algebras.
We show that free objects on sets do not exist in the category $bal$ of bounded archimedean $\ell$-algebras. On the other hand, we introduce the category of weighted sets and prove that free objects on weighted sets do exist in $bal$. We conclude by discussing several consequences of this result.
We prove that the variety of nuclear implicative semilattices is locally finite, thus generalizing Diego's Theorem. The key ingredients of our proof include the coloring technique and construction of universal models from modal logic. For this we develop duality theory for finite nuclear implicative semilattices, generalizing Köhler duality. We prove that our main result remains true for bounded nuclear implicative semilattices, give an alternative proof of Diego's Theorem, and provide an explicit description of the free cyclic nuclear implicative semilattice.
The variety of Brouwerian semilattices is amalgamable and locally finite, hence by well-known results due to W. H. Wheeler, it has a model completion (whose models are the existentially closed structures). In this paper, we supply for such a model completion a finite and rather simple axiomatization.