d-Frames as algebraic duals of bitopological spaces

Achim Jung and Drew Moshier developed a Stone-type duality theory for bitopological spaces, amongst others, as a practical tool for solving a particular problem in the theory of stably compact spaces. By doing so they discovered that the duality of bitopological spaces and their algebraic counterparts, called d-frames, covers several of the known dualities. In this thesis we aim to take Jung's and Moshier's work as a starting point and fill in some of the missing aspects of the theory. In particular, we investigate basic categorical properties of d-frames, we give a Vietoris construction for d-frames which generalises the corresponding known Vietoris constructions for other categories, and we investigate the connection between bispaces and a paraconsistent logic and then develop a suitable (geometric) logic for d-frames

Duality and canonical extensions for stably compact spaces

We construct a canonical extension for strong proximity lattices in order to give an algebraic, point-free description of a finitary duality for stably compact spaces. In this setting not only morphisms, but also objects may have distinct pi- and sigma-extensions.Comment: 29 pages, 1 figur

Bitopology and four-valued logic

AbstractBilattices and d-frames are two different kinds of structures with a four-valued interpretation. Whereas d-frames were introduced with their topological semantics in mind, the theory of bilattices has a closer connection with logic. We consider a common generalisation of both structures and show that this not only still has a clear bitopological semantics, but that it also preserves most of the original bilattice logic. Moreover, we also obtain a new bitopological interpretation for the connectives of four-valued logic

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

Pointfree bispaces and pointfree bisubspaces

This thesis is concerned with the study of pointfree bispaces, and in particular with the pointfree notion of inclusion of bisubspaces. We mostly work in the context of d-frames. We study quotients of d-frames as pointfree analogues of the topological notion of bisubspace. We show that for every d-frame L there is a d-frame A(L) such that it plays the role of the assembly of a frame, in the sense that it has the analogue of the universal property of the assembly and that its spectrum is a bitopological version of the Skula space of the bispace dpt(L), the spectrum of L. Furthermore, we show that this bitopological version of the Skula space of dpt(L) is the coarsest topology in which the d-sober bisubspaces of dpt(L) are closed. We also show that there are two free constructions in the category of d-frames Act(L) and A_(L), such that they represent two variations of the bitopological version of the Skula topology. In particular, we show that in dpt(Act) the positive closed sets are exactly those d-sober subspaces of dpt(L) that are spectra of quotients coming from an increase in the con component, and that the negative closed ones are those that come from increases in the tot component. For dpt(A_(L)), we show that the positive closed sets are exactly those bisubspaces of dpt(L) that are spectra of quotients coming from a quotient of L+, and that the negative closed sets come in the same way from quotients of

Bi-intermediate logics of trees and co-trees

A bi-Heyting algebra validates the G\"odel-Dummett axiom $(p\to q)\vee (q\to p)$ iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-G\"odel algebras and form a variety that algebraizes the extension $\mathsf{bi}$-$\mathsf{LC}$ of bi-intuitionistic logic axiomatized by the G\"odel-Dummett axiom. In this paper we initiate the study of the lattice $\Lambda(\mathsf{bi}$-$\mathsf{LC})$ of extensions of $\mathsf{bi}$-$\mathsf{LC}$. We develop the methods of Jankov-style formulas for bi-G\"odel algebras and use them to prove that there are exactly continuum many extensions of $\mathsf{bi}$-$\mathsf{LC}$. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of $\mathsf{bi}$-$\mathsf{LC}$. We introduce a sequence of co-trees, called the finite combs, and show that a logic in $\mathsf{bi}$-$\mathsf{LC}$ is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest non-locally tabular extension of $\mathsf{bi}$-$\mathsf{LC}$ and consequently, a unique pre-locally tabular extension of $\mathsf{bi}$-$\mathsf{LC}$. These results contrast with the case of the intermediate logic axiomatized by the G\"odel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular

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