Dual attachment pairs in categorically-algebraic topology

[EN] The paper is a continuation of our study on developing a new approach to (lattice-valued) topological structures, which relies on category theory and universal algebra, and which is called categorically-algebraic (catalg) topology. The new framework is used to build a topological setting, based in a catalg extension of the set-theoretic membership relation "e" called dual attachment, thereby dualizing the notion of attachment introduced by the authors earlier. Following the recent interest of the fuzzy community in topological systems of S. Vickers, we clarify completely relationships between these structures and (dual) attachment, showing that unlike the former, the latter have no inherent topology, but are capable of providing a natural transformation between two topological theories. We also outline a more general setting for developing the attachment theory, motivated by the concept of (L,M)-fuzzy topological space of T. Kubiak and A. Sostak.This research was partially supported by the ESF Project of the University of Latvia No. 2009/0223/1DP/1.1.1.2.0/09/APIA/VIAA/008.Frascella, A.; Guido, C.; Solovyov, SA. (2011). Dual attachment pairs in categorically-algebraic topology. Applied General Topology. 12(2):101-134. doi:10.4995/agt.2011.1646.SWORD10113412

A study of localic subspaces, separation, and variants of normality and their duals.

198 p.As in classical topology, in localic topology one often needs to restrict to locales satisfyinga certain degree of separation. In fact, the study of separation in the category of localesconstitutes a non-trivial and important piece of the theory. For instance, it is sometimesimpossible to give an exact counterpart of a classical axiom, while other times a singleproperty for spaces yields multiple non-equivalent localic versions.The main goal of this thesis is to investigate several classes of separated locales and theirconnections with different classes of sublocales, that is, the regular subobjects in the categoryof locales.In particular, we introduce a new diagonal separation and show that it is, in a certainsense, dual to Isbell¿s (strong) Hausdorff property. The duality between suplattices andpreframes, and that between normality and extremal disconnectedness, turn out to be ofspecial interest in this context.Regarding higher separation, we introduce cardinal generalizations of normality andtheir duals (e.g., properties concerning extensions of disjoint families of cozero elements),and give characterizations via suitable insertion or extension results.The lower separation property known as the TD-axiom, also plays an important role inthe thesis. Namely, we investigate the TD-duality between the category of TD-spaces and acertain (non-full) subcategory of the category of locales, identifying the regular subobjects inthe localic side, and provide several applications in point-free topology.Tal como na topologia clássica, também na topologia dos locales (reticulados locais) éfrequente termos que nos restringir a locales que satisfaçam um certo grau de separação.De facto, o estudo de axiomas de separação na categoria dos locales constitui um aspectonão trivial e relevante da teoria. Por exemplo, em alguns casos é impossível termos acontrapartida exacta de um axioma clássico, enquanto noutros casos uma única propriedadepara espaços topológicos produz, na categoria dos locales, diversas versões não equivalentesentre si.O objectivo principal desta tese é investigar várias classes de locales separados e suasconexões com diferentes classes de sublocales (os subobjetos regulares na categoria doslocales).Em particular, introduzimos uma nova propriedade de separação diagonal e mostramosque se trata, em certo sentido, de uma propriedade dual do axioma (forte) de Hausdorffintroduzido por Isbell. As dualidades entre semi-reticulados e reticulados pré-locais, e entrenormalidade e desconexão extrema, acabam por ter um papel relevante neste contexto.Relativamente a axiomas de separação fortes, introduzimos generalizações de normalidade,em função de um cardinal arbitrário, e suas duais (por exemplo, propriedadesenvolvendo extensões de famílias disjuntas de elementos co-zero), e apresentamos caracterizaçõesem termos de propriedades de inserção ou extensão de funções.O axioma TD, uma propriedade de separação muito fraca, também desempenha umpapel importante nesta tese. Especificamente, investigamos a dualidade TD entre a categoriados espaços topológicos TD e uma determinada subcategoria (não plena) da categoria doslocales, identificando os subobjetos regulares na subcategoria de locales, e apresentamosvárias aplicações à topologia sem pontos.Tal y como ocurre en topología clásica, en topología locálica frecuentemente uno tiene querestringir su atención a locales que cumplen cierto grado de separación. De hecho, el estudiode la separación en la categoría de locales es un aspecto no trivial y relevante de la teoría. Enalgunos casos, es imposible dar una contrapartida exacta a un axioma clásico, mientras queen otros casos, una sola propiedad produce multitud de versiones locálicas no equivalentesentre sí.El principal objetivo de esta tesis es investigar varias clases de locales separados y susrelaciones con diferentes clases de sublocales, esto es, los subobjetos regulares en la categoríade locales.En particular, introducimos una nueva separación diagonal, y probamos que es, en ciertosentido, dual al axioma Hausdorff (fuerte) de Isbell. En este contexto, la dualidad entreretículos completos y premarcos, y aquella entre la normalidad y la desconexión extremaresultan ser de especial interés.En cuanto a la separación más fuerte, introducimos generalizaciones cardinales de lanormalidad y sus duales (por ejemplo, propiedades que consisten en la extensión de familiasdisjuntas de elementos cozero), y damos caracterizaciones de las mismas en términos deteoremas de extensión o inserción.Ciertas propiedades de separación más débiles, especialmente el axioma TD, tambiéndesempeñan un papel importante en esta tesis. Específicamente, investigamos la dualidad TDentre la categoría de espacios topológicos TD y cierta subcategoría (no plena) de la categoríade locales, identificando los subobjetos regulares en la categoría de locales, y proporcionamosalgunas aplicaciones en la topología sin puntos

Mathematics in the context of fuzzy sets: basic ideas, concepts, and some remarks on the history and recent trends of development

The main aim of this paper is to discuss the basic ideas and concepts of the so called ‘Fuzzy Mathematics’ and to give a brief survey of the history and of some trends in recent development of mathematics and its applications in the context of fuzzy sets. As a potential reader we imagine a mathematician, who is not working in the field of ‘fuzzy mathematics’, but wishes to have some idea about this vast field in modern science

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

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

Foundations of Quantum Theory: From Classical Concepts to Operator Algebras

Quantum physics; Mathematical physics; Matrix theory; Algebr