Heyting frames and Esakia duality

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

On the free implicative semilattice extension of a Hilbert algebra

Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.Fil: Celani, Sergio Arturo. Universidad Nacional del Centro de la Provincia de Buenos Aires; Argentina. Universidad Central de Barcelona; España. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Tandil; ArgentinaFil: Jansana, Ramon. Universidad Nacional del Centro de la Provincia de Buenos Aires; Argentina. Universidad de Barcelona; Españ

Positive Modal Logic Beyond Distributivity

We develop a duality for (modal) lattices that need not be distributive, and use it to study positive (modal) logic beyond distributivity, which we call weak positive (modal) logic. This duality builds on the Hofmann, Mislove and Stralka duality for meet-semilattices. We introduce the notion of $\Pi_1$-persistence and show that every weak positive modal logic is $\Pi_1$-persistent. This approach leads to a new relational semantics for weak positive modal logic, for which we prove an analogue of Sahlqvist correspondence result

Modal meet-implication logic

We extend the meet-implication fragment of propositional intuitionistic logic with a meet-preserving modality. We give semantics based on semilattices and a duality result with a suitable notion of descriptive frame. As a consequence we obtain completeness and identify a common (modal) fragment of a large class of modal intuitionistic logics. We recognise this logic as a dialgebraic logic, and as a consequence obtain expressivity-somewhere-else. Within the dialgebraic framework, we then investigate the extension of the meet-implication fragment of propositional intuitionistic logic with a monotone modality and prove completeness and expressivity-somewhere-else for it

Priestley-Stone Duality for Subbases of Stably Locally Compact Spaces

We extend the classic Priestley-Stone duality to a Wallman-like duality for subbases of general stably locally compact spaces. As a corollary, we show that any locally compact T_0 space X has a unique minimal 'stabilisation', i.e. a stably locally compact space containing X as a patch-dense subspace, which is moreover functorial with respect to proper maps

On homomorphic images and the free distributive lattice extension of a distributive nearlattice

In this paper we will introduce N-Vietoris families and prove that homomorphic images of distributive nearlattices are dually characterized by N-Vietoris families. We also show a topological approach of the existence of the free distributive lattice extension of a distributive nearlattice.Fil: Celani, Sergio Arturo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Tandil; Argentina. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Departamento de Matemática; ArgentinaFil: Calomino, Ismael Maria. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Tandil; Argentina. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Departamento de Matemática; Argentin

Dualities in modal logic

Categorical dualities are an important tool in the study of (modal) logics. They offer conceptual understanding and enable the transfer of results between the different semantics of a logic. As such, they play a central role in the proofs of completeness theorems, Sahlqvist theorems and Goldblatt-Thomason theorems. A common way to obtain dualities is by extending existing ones. For example, Jonsson-Tarski duality is an extension of Stone duality. A convenient formalism to carry out such extensions is given by the dual categorical notions of algebras and coalgebras. Intuitively, these allow one to isolate the new part of a duality from the existing part. In this thesis we will derive both existing and new dualities via this route, and we show how to use the dualities to investigate logics. However, not all (modal logical) paradigms fit the (co)algebraic perspective. In particular, modal intuitionistic logics do not enjoy a coalgebraic treatment, and there is a general lack of duality results for them. To remedy this, we use a generalisation of both algebras and coalgebras called dialgebras. Guided by the research field of coalgebraic logic, we introduce the framework of dialgebraic logic. We show how a large class of modal intuitionistic logics can be modelled as dialgebraic logics and we prove dualities for them. We use the dialgebraic framework to prove general completeness, Hennessy-Milner, representation and Goldblatt-Thomason theorems, and instantiate this to a wide variety of modal intuitionistic logics. Additionally, we use the dialgebraic perspective to investigate modal extensions of the meet-implication fragment of intuitionistic logic. We instantiate general dialgebraic results, and describe how modal meet-implication logics relate to modal intuitionistic logics

Representations and Completions for Ordered Algebraic Structures

The primary concerns of this thesis are completions and representations for various classes of poset expansion, and a recurring theme will be that of axiomatizability. By a representation we mean something similar to the Stone representation whereby a Boolean algebra can be homomorphically embedded into a field of sets. So, in general we are interested in order embedding posets into fields of sets in such a way that existing meets and joins are interpreted naturally as set theoretic intersections and unions respectively. Our contributions in this area are an investigation into the ostensibly second order property of whether a poset can be order embedded into a field of sets in such a way that arbitrary meets and/or joins are interpreted as set theoretic intersections and/or unions respectively. Among other things we show that unlike Boolean algebras, which have such a ‘complete’ representation if and only if they are atomic, the classes of bounded, distributive lattices and posets with complete representations have no first order axiomatizations (though they are pseudoelementary). We also show that the class of posets with representations preserving arbitrary joins is pseudoelementary but not elementary (a dual result also holds). We discuss various completions relating to the canonical extension, whose classical construction is related to the Stone representation. We claim some new results on the structure of classes of poset meet-completions which preserve particular sets of meets, in particular that they form a weakly upper semimodular lattice. We make explicit the construction of \Delta_{1}-completions using a two stage process involving meet- and join-completions. Linking our twin topics we discuss canonicity for the representation classes we deal with, and by building representations using a meet-completion construction as a base we show that the class of representable ordered domain algebras is finitely axiomatizable. Our method has the advantage of representing finite algebras over finite bases