A Generalization of the Stone Duality Theorem

We prove a new duality theorem for the category of precontact algebras which implies the Stone Duality Theorem, its connected version obtained in arXiv:1508.02220v3, 1-44 (to appear in Topology Appl.), the recent duality theorems of Bezhanishvili, G., Bezhanishvili, N., Sourabh, S., Venema, Y. and Goldblatt, R. and Grice, M, and some new duality theorems for the category of contact algebras and for the category of complete contact algebras.Comment: 29 pages. arXiv admin note: substantial text overlap with arXiv:1508.0222

Some Isomorphism Theorems for MVD-algebras

In three recaent papers of G. Dimov, many Stone-type duality theorems for the category of locally compact Hausdorff spaces and continuous maps and some of its subcategories were proved. The dual objects in all these theorems are the local contact algebras. In a paper of D. Vakarelov, G. Dimov, I. Duntsch, and B. Bennett, the notion of an MVD-algebra was introduced and it was shown that it is equivalent to the notion of a local contact algebra. In this paper we express the duality theorems mentioned above in a new form using MVD-algebras and appropriate morphisms between them instead of local contact algebras and the respective morphisms.Comment: 18 pages. arXiv admin note: text overlap with arXiv:0709.4495 by other author

A new duality theorem for locally compact spaces

In 1962, H. de Vries proved a duality theorem for the category {\bf HC} of compact Hausdorff spaces and continuous maps. The composition of the morphisms of the dual category obtained by him differs from the set-theoretic one. Here we obtain a new category dual to the category {\bf HLC} of locally compact Hausdorff spaces and continuous maps for which the composition of the morphisms is a natural one but the morphisms are multi-valued maps.Comment: 18 page

On the Structure and Function of Scientific Perspectivism in Quantum Mechanics

Contemporary scientific perspectivism is re-evaluated and extended to a comprehensive perspectivist methodology and 'mediated' realistic epistemology, especially, with reference to quantum mechanics. In the present study, this is realized by representing categorically the global structure of a quantum algebra of events in terms of structured multitudes of interrelated local Boolean frames, realized as suitable perspectives or contexts for measuring physical quantities. The philosophical meaning of the proposed approach implies that the quantum world can be consistently approached and comprehended through a multilevel structure of locally variable perspectives, which interlock, in a category-theoretical environment, to form a coherent picture of the whole in a nontrivial way. Thus, in contrast to a panoptical "view from nowhere" of the classical paradigm, quantum theory acknowledges in an essential way a perspectival/contextual character of scientific knowledge.Comment: 16 pages, 3 figure

Some Generalizations of Fedorchuk Duality Theorem -- I

Generalizing Duality Theorem of V. V. Fedorchuk, we prove Stone-type duality theorems for the following four categories: all of them have as objects the locally compact Hausdorff spaces, and their morphisms are, respectively, the continuous skeletal maps, the quasi-open perfect maps, the open maps, the open perfect maps. In particular, a Stone-type duality theorem for the category of all compact Hausdorff spaces and all open maps between them is proved. We also obtain equivalence theorems for these four categories. The versions of these theorems for the full subcategories of these categories having as objects all locally compact connected Hausdorff spaces are formulated as well.Comment: 28 page

Point-free theories of space and time

The paper is in the field of Region Based Theory of Space (RBTS), sometimes called mereotopology. RBTS is a kind of point-free theory of space based on the notion of region. Its origin goes back to some ideas of Whitehead, De Laguna and Tarski to build the theory of space without the use of the notion of point. More information on RBTS and mereotopology can be found, for instance, in \cite{Vak2007}. Contact algebras present an algebraic formulation of RBTS and in fact give axiomatizations of the Boolean algebras of regular closed sets of some classes of topological spaces with an additional relation of contact. An exhaustive study of this theory is given in \cite{DiVak2006}. Dynamic contact algebra (DCA) \cite{Vak2014} (see also \cite{Vak2010,Vak2012}) introduced by the present author, is a generalization of contact algebra studying regions changing in time and presents a formal explication of Whitehead's ideas of integrated point-free theory of space and time. DCA is an abstraction of a special \emph{dynamic model of space}, called also \emph{snapshot} or \emph{cinematographic} model and the paper \cite{Vak2014} contains the expected representation theorem with respect to such models. In the present paper we introduce a new version of DCA which is a simplified version of the definition from \cite{Vak2014} and similar to that of \cite{Vak2012}. The aim is to use this version as a representative example of a DCA and to develop for this example not only the snapshot models but also topological models and the expected topological duality theory, generalizing in a certain sense the well known Stone duality for Boolean algebras. Abstract topological models of DCAs present a new view on the nature of space and time and show what happens if we are abstracting from their metric properties.Comment: 89 page

Algebraic analysis of temporal and topological finite variable fragments, using cylindric modal algebras

We study what we call topological cylindric algebras and tense cylindric algebras defined for every ordinal $\alpha$. The former are cylindric algebras of dimension $\alpha$ expanded with $\sf S4$ modalities indexed by $\alpha$. The semantics of representable topological algebras is induced by the interior operation relative to a topology defined on their bases. Tense cylindric algebras are cylindric algebras expanded by the modalities $F$(future) and $P$ (past) algebraising predicate temporal logic. We show for both tense and topological cylindric algebras of finite dimension $n>2$ that infinitely many varieties containing and including the variety of representable algebras of dimension $n$ are not atom canonical. We show that any class containing the class of completely representable algebras having a weak neat embedding property is not elementary. From these two results we draw the same conclusion on omitting types for finite variable fragments of predicate topologic and temporal logic. We show that the usual version of the omitting types theorem restricted to such fragments when the number of variables is $>2$ fails dramatically even if we considerably broaden the class of models permitted to omit a single non principal type in countable atomic theories, namely, the non-principal type consting of co atoms.Comment: arXiv admin note: substantial text overlap with arXiv:1308.6165, arXiv:1307.1016, arXiv:1309.0681, arXiv:1307.4298, arXiv:1401.1103, arXiv:1401.115

Representation theorems for extended contact algebras based on equivalence relations

The aim of this paper is to give new representation theorems for extended contact algebras. These representation theorems are based on equivalence relations

A Bridge Between Q-Worlds

Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding quantum mechanics by reformulating parts of the theory inside of non-classical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, `Q-worlds'. Here, we provide a unifying framework that allows us to (i) better understand the relationship between different Q-worlds, and (ii) define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory.Comment: v2: 40 pages, latex. Typos are corrected and 4 references are added. Readability of some proofs are improved with intermediate steps in the formulae. Discussions on weakly self-adjoint operators are moved to the forthcoming pape

A system of relational syllogistic incorporating full Boolean reasoning

We present a system of relational syllogistic, based on classical propositional logic, having primitives of the following form: Some A are R-related to some B; Some A are R-related to all B; All A are R-related to some B; All A are R-related to all B. Such primitives formalize sentences from natural language like `All students read some textbooks'. Here A and B denote arbitrary sets (of objects), and R denotes an arbitrary binary relation between objects. The language of the logic contains only variables denoting sets, determining the class of set terms, and variables denoting binary relations between objects, determining the class of relational terms. Both classes of terms are closed under the standard Boolean operations. The set of relational terms is also closed under taking the converse of a relation. The results of the paper are the completeness theorem with respect to the intended semantics and the computational complexity of the satisfiability problem.Comment: Available at http://link.springer.com/article/10.1007/s10849-012-9165-