Many Valued Topology and its Applications

Höhle, Ulrich

doi:10.1007/978-1-4615-1617-0

Cited by 111 publications

(113 citation statements)

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“…It is quite different from the situation in "Fuzzy Topology", which is a very well developed area of theoretical mathematics with many fundamental concepts, constructions and results, and which has important applications to other fields of mathematics, see e.g. [27], [19].…”

Section: Bornologies and Bornological Spacesmentioning

confidence: 98%

“…As an additional incentive to develop our approach in the general framework of many-valued sets instead of ordinary sets, we consider the role of many-valued sets in the theory of fuzzy topologies: just in this framework many interesting results and examples are obtained, in particular applications related to other fields of theoretical mathematics, see e.g. [19].…”

Section: (2mb) If a ⊆ B Then B(a) ≥ B(b); (3mb) For Every A B ⊆ X Itmentioning

confidence: 99%

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Bornological structures on many-valued sets

Šostak¹,

Uljane²

2017

Rad Hrvatske akademije znanosti i umjetnosti

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Abstract. We introduce an approach to the concept of bornology in the framework of many-valued mathematical structures and develop the basics of the theory of many-valued bornological spaces and initiate the study of the category of many-valued bornological spaces and appropriately defined bounded "mappings" of such spaces. A scheme for constructing many-valued bornologies with prescribed properties is worked out. In particular, this scheme allows to extend an ordinary bornology of a metric space to a many-valued bornology on it.In the academic year 1978/79 I was lucky to get a post-doc research position at the University of Zagreb and to spend this time working in collaboration with professor Sibe Mardešić in shape theory. It was a real pleasure to work together and to communicate generally with this outstanding mathematician, talented teacher and a very kind and nice man. Unfortunately, at present I am quite far from shape theory. Therefore the paper I would like to dedicate to Sibe Mardešić, written together with my former PhD student Ingrīda Uljane, is in the field of so called many-valued mathematical structures -the area of my main mathematical interests at present.

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Section: Bornologies and Bornological Spacesmentioning

confidence: 98%

Section: (2mb) If a ⊆ B Then B(a) ≥ B(b); (3mb) For Every A B ⊆ X Itmentioning

confidence: 99%

Bornological structures on many-valued sets

Šostak¹,

Uljane²

2017

Rad Hrvatske akademije znanosti i umjetnosti

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“…To show its value and usefulness, we gathered some familiar cases of C-M-L-Top in the following example. [14].…”

Section: And Whose Morphismsmentioning

confidence: 99%

“…The famous adjunction Ω ⊣ P t between the category Top of topological spaces and the opposite Loc of the category Frm of frames [16,17,19], known as Papert-Papert-Isbell adjunction [24], and its various generalizations in fuzzy set theory have received much attention during the last three decades [2,3,4,5,14,17,20,23,24]. Also see [4,24] for many other references not included in this paper.…”

Section: Introductionmentioning

confidence: 99%

Applications of Categorical Fixed-basis Fuzzy Topological Spaces to A-valued Spaces

Demirci¹

2013

Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technology

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As an application of the dual equivalence between the category of L-spatial C-objects and the category of L-sober C-M-L-spaces, it is shown in this paper that for a fixed augmented partially ordered set A, there exists a dual equivalence between the category of A-spatial augmented partially ordered sets and the category of A-sober A-valued spaces. Then, with regard to this duality, for a fixed (Z 1 , Z 2 )-complete partially ordered set L, we establish a dual equivalence between the category of L-spatial (Z 1 , Z 2 )-complete partially ordered sets and the category of L-sober L-valued Q-spaces.

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“…Recently, quantales have arisen in an analysis of the semantics of linear logic systems developed by Girard [5], which supports part of foundation of theoretic computer science. Höhle [7][8][9] developed the algebraic structures and many valued topologies in a sense of quantales and cqm-lattices. Bělohlávek [1][2][3] investigate the properties of fuzzy relations and similarities on a residual lattice.…”

Section: Introductionmentioning

confidence: 99%