Interweaving algebra and topology: Lattice-valued topological systems

Denniston, Jeffrey T.; Melton, Austin; Rodabaugh, S. E.

doi:10.1016/j.fss.2011.07.001

Cited by 32 publications

(20 citation statements)

References 15 publications

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“…(2) If A = B = Frm, then the category Loc-TopSys is the category of lattice-valued topological systems of Denniston et al [8,9].…”

Section: Example 24mentioning

confidence: 99%

“…The authors brought their theory into maturity in [9], where they considered its applications to both lattice-valued variable-basis topology of Rodabaugh [4] and ( , )-fuzzy topology of Kubiak anď Sostak [10]. Later on, they introduced a particular instance of their concept called interchange system [11], which was motivated by certain aspects of program semantics (the socalled predicate transformers) initiated by Dijkstra [12].…”

Section: Introductionmentioning

confidence: 99%

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Lattice-Valued Topological Systems as a Framework for Lattice-Valued Formal Concept Analysis

Solovyov

2013

Journal of Mathematics

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Recently, Denniston, Melton, and Rodabaugh presented a new categorical outlook on a certain lattice-valued extension of Formal Concept Analysis (FCA) of Ganter and Wille; their outlook was based on the notion of lattice-valued interchange system and a category of Galois connections. This paper extends the approach of Denniston et al. clarifying the relationships between Chu spaces of Pratt, many-valued formal contexts of FCA, lattice-valued interchange systems, and Galois connections.

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“…(2) If A = B = Frm, then the category Loc-TopSys is the category of lattice-valued topological systems of Denniston et al [8,9].…”

Section: Example 24mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lattice-Valued Topological Systems as a Framework for Lattice-Valued Formal Concept Analysis

Solovyov

2013

Journal of Mathematics

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“…This time, it is the concept of topological system introduced by S. Vickers [75] as a common framework for incorporating both topological spaces and their underlying algebraic structures -locales [35], thereby trying to merge point-set and pointless topology. Recently, the notion was successfully extended to include the case of lattice-valued topologies, the most significant results in the field achieved by J. T. Denniston, A. Melton, S. E. Rodabaugh [11,12,13,14], C. Guido [25,26] and S. Solovyov [66,74]. For the sake of shortness, the category (LoA, LoA)-TopSys is denoted LoA-TopSys, whereas the category (S A , LoA)-TopSys is denoted A-TopSys.…”

Section: Lob-topmentioning

confidence: 99%

“…)), where CLat is the variety of complete lattices and CDCLat is its subcategory of completely distributive lattices, provides a categorical accommodation of the theory of (L,M )-fuzzy topological spaces of T. Kubiak and A.Šostak [40]. (2) LTop((R 3 , Frm, Frm)) is isomorphic to the category Loc-F 2 Top of (L, M )-fuzzy topological spaces of J. T. Denniston, A. Melton and S. E. Rodabaugh [11], which was introduced as a variable-basis counterpart of the above-mentioned approach of T. Kubiak and A.Šostak. In view of the above-mentioned remarks, it seems natural to consider the notion of attachment in the more general lattice-valued framework, and, therefore, one can postulate the following open problem.…”

Section: Conclusion: Lattice-valued Categorically-algebraic Topologymentioning

confidence: 99%

Dual attachment pairs in categorically-algebraic topology

2013

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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 "∈" 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.Šostak.

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“…More precisely, given a (fibre-small) topological category A, there exists a topological category B, which contains A as a full concretely coreflective subcategory, and which can be considered as a fuzzification of A. In particular, the classical category Top of topological spaces (in the role of the above category A) provides an analogue of the original spaces of U. Höhle, the category L-Top of Chang-Goguen L-topological spaces provides a (partially, variable-basis) analogue of the spaces of Kubiak-Šostak, whereas the category Loc-Top of [3] gives an analogue of the variable-basis lattice-valued topology in the sense of Höhle-Kubiak-Šostak, which has been recently developed by J. T. Denniston, A. Melton and S. E. Rodabaugh [4]. …”

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confidence: 99%

On fuzzification of topological categories

Solovyov¹

EPiC Series in Computing

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Following the point-set lattice-theoretic (poslat) approach to topology of S. E. Rodabaugh [11], we introduced a more general way to do categorical (fuzzy) topology under the name of categorically-algebraic (catalg) topology [12], which unified many lattice-valued topological settings, and provided the means of intercommunication between different such frameworks. Moreover, motivated by the results of universal topology of H. Herrlich [7], we showed in [13] that a concrete category is fibre-small and topological if and only if it is concretely isomorphic to a subcategory of a category of catalg topological structures, which is definable by topological co-axioms. This achievement relies on the machinery of topological theories of O. Wyler [18,19], and eventually amounts to the meta-mathematical statement that the whole theory of (fibresmall) topological categories (including, e.g., the categories of poslat topological spaces, shown to be topological over their ground categories in a series of papers of S. E. Rodabaugh) can be done in the framework of catalg topology (a similar but by far more moderate and not that well stated claim was vaguely hinted at in [11] with respect to the poslat topology). The purpose of this talk (which is a shortened version of [14]) is to show a particular development of fuzzy topology in the setting of the catalg one, thereby illustrating the convenient tools of the latter.There currently exist two popular (in the fuzzy community) approaches to lattice-valued topology (which by no means are the only available). The first one, initiated by C. L. Chang [1] and extended later on by J. A. Goguen [6], defines a many-valued topology on a set X as a subset τ of an L-powerset L X (for some lattice-theoretic structure L, e.g., a quantale), which is closed under arbitrary joins and finite meets (or, possibly, quantale multiplication). The second approach, started by U. Höhle [8] and later on generalized independently by T. Kubiak [9] and A.Šostak [17], defines a lattice-valued topology on a set X as a map T : L X − → M (which employs an additional lattice-theoretic structure M ) satisfying the requirements i∈I T (α i ) T ( i∈I α i ) for every index set I, and ∧ j∈J T (α j ) T (∧ j∈J α j ) for every finite index set J, which provide lattice-valued analogues of the above-mentioned closure of topology under arbitrary joins and finite meets. In [15,16], we presented a convenient catalg framework for doing fuzzy topology in the sense of Höhle-Kubiak-Šostak, extending for that purpose the theory of latticevalued algebras of A. Di Nola and G. Gerla [5]. In this talk, we show that the approach to topology of Höhle-Kubiak-Šostak is just an instance of the machinery, which has been called by us the general fuzzification procedure for topological categories. More precisely, given a (fibre-small) topological category A, there exists a topological category B, which contains A as a full concretely coreflective subcategory, and which can be considered as a fuzzification of A. In particular, the classical cat...

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Interweaving algebra and topology: Lattice-valued topological systems

Cited by 32 publications

References 15 publications

Lattice-Valued Topological Systems as a Framework for Lattice-Valued Formal Concept Analysis

Lattice-Valued Topological Systems as a Framework for Lattice-Valued Formal Concept Analysis

Dual attachment pairs in categorically-algebraic topology

On fuzzification of topological categories

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