| 1. |
- Eklund, Patrik, 1958-, et al.
(author)
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Basic notions for fuzzy topology, I
- 1988
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In: Fuzzy sets and systems (Print). - : Elsevier BV. - 0165-0114 .- 1872-6801. ; 26:3, s. 333-356
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Journal article (peer-reviewed)
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| 2. |
- Eklund, Patrik, 1958-, et al.
(author)
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Basic notions for fuzzy topology, II
- 1988
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In: Fuzzy sets and systems (Print). - : Elsevier BV. - 0165-0114 .- 1872-6801. ; 27:2, s. 171-195
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Journal article (peer-reviewed)
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| 3. |
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| 4. |
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| 5. |
- Eklund, Patrik, 1958-, et al.
(author)
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Fuzzy Filter Functors and Convergence
- 1991
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In: Applications of category theory to fuzzy subsets. - : Kluwer Academic Publishers. - 0792315111 ; , s. 110-136
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Book chapter (peer-reviewed)
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| 6. |
- Eklund, Patrik, 1958-, et al.
(author)
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Generalized Cauchy spaces
- 1990
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In: Mathematische Nachrichten. - : Wiley-Blackwell. - 0025-584X .- 1522-2616. ; 147:1, s. 219-233
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Journal article (peer-reviewed)abstract
- In this paper a unified theory of Cauchy spaces is presented including the classical cases of filter and sequence Cauchy spaces. To by-pass a lattice-theoretical barrier the notion of Urysohn modification of a functor is introduced. Employing this notion for many types of generalized Cauchy spaces a completion method is given.
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| 7. |
- Eklund, Patrik, 1958-, et al.
(author)
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Partially ordered monads for monadic topologies, Kleene algebras and rough sets
- 2009
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In: Electronic Notes in Theoretical Computer Science. - : Elsevier. - 1571-0661. ; 225, s. 67-81
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Journal article (peer-reviewed)abstract
- In this paper we will show that partially ordered monads contain sufficient structure for modelling monadic topologies, rough sets and Kleene algebras. Convergence represented by extension structures over partially ordered monads includes notions of regularity and compactness. A compactification theory can be developed. Rough sets [Z. Pawlak, Rough sets, Int. J. Computer and Information Sciences 5 (1982) 341356] are modelled in a generalized setting with set functors. Further, we show how partially ordered monads can be used in order to obtain monad based examples of Kleene algebras building upon a wide range of set functors far beyond just strings [S. C. Kleene, Representation of events in nerve nets and finite automata, In: Automata Studies (Eds. C. E. Shannon, J. McCarthy), Princeton University Press, 1956, 3-41] and relations [A. Tarski, On the calculus of relations, J. Symbolic Logic 6 (1941), 65-106].
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| 8. |
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| 9. |
- Gähler, Werner, et al.
(author)
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Extension structures and compactifications
- 2001
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In: Proceedings Joint 9th IFSA World Congress and 20th NAFIPS International Conference (Vol. 5). - 0780370783 ; , s. 2940-2945
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Conference paper (peer-reviewed)abstract
- Basic results on compactifications are presented applying the notion of extension structure. Eachextension structure has a canonical completion. The related completion constructions can be applied, for instance, for generating completion theorems in algebra, lattice theory and general topology, in particular they lead to a universal completion for Cauchy-spaces in the fuzzy filter case. Sincecompactifications can be identified with special Cauchy-completions, even different types ofcompactifications can be generated. Among others, we present new results on the Richardson compactification in the fuzzy filter case applying new results on fuzzy filters. This type of compactification was treated previously by the authors (1993)
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| 10. |
- Gähler, Werner, et al.
(author)
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Extension structures and compactifications
- 2000
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Conference paper (peer-reviewed)abstract
- The notion of extension structure [2] depends on a fixed set functor j: \sf SET --> \sf SET. We therefore also speak on a j-extension structure. This notion contains important structural properties of Cauchy structures and is basically for a general completion theory. Let X be a set. By a j-extension structure on X we mean a triple (S, T, ~ ) consisting of two sets S subset or equal jX and T subset or equal jX ×X and an equivalence relation ~ on S such that, writing M \usebox\ko x instead of (M, x) in T, we have that (1) M \usebox\ko x implies M in S, (2) M \usebox\ko x and M ~ N imply N \usebox\ko x, and (3) from M \usebox\ko x and N \usebox\ko x it follows M ~ N.A j-extension structure (S, T, ~ ) and also the related j-extension space (X, (S, T, ~ )) is said to be separated provided that M \usebox\ko x and M\usebox\ko y imply x = y, and they are said to be complete provided that for each M in S there is an element x of X such thatM \usebox\ko x holds.We have that the category of all separated and complete j-extension spaces is an epireflective subcategory of the category of all separated j-extension spaces. As is shown in [3], the related completion construction can be applied for generating completion theorems in algebra, lattice theory and general topology, in particular they lead to a universal completion for Cauchy-spaces in the fuzzy filter case. Since compactifications can be identified with special Cauchy-completions, even different types of compactifications can be generated.Among others, we present in this paper new results on the Richardson compactification for the fuzzy filter case applying new results on fuzzy filters. Note that this type of compactification was already treated in [1]. For a fixed non-degenerate infinitely distributive complete lattice L, here we mean by a fuzzy filter a mapping M: LX --> L such that M([`0]) = 0 and M([`1]) = 1 for the constant fuzzy sets [`0], [`1] in LX and M(f /\ g) = M(f) /\ M(g) for all fuzzy sets f, g in LX.A fuzzy filter M is said to be bounded in case M(f) <= supf holds for all f in LX. A fuzzy filter M will be called distinguished provided that M(f \/ g) = M(f) \/ M(g) holds for all f, g in LX. Moreover, M will be called balanced provided that a distinguished fuzzy filter finer thanM exists.In the general fuzzy filter case under the zero-condition of L (\alpha > 0, \beta > 0 ===> \alpha /\ \beta > 0) all fuzzy filters are balanced. In this case the Richardson compactification exists.In the case of bounded fuzzy filters there may be fuzzy filters which are not balanced. If in this case we restrict us to balanced fuzzy filters, then the Richardson compactification also exists.[1] P. Eklund, W. Gähler, Completions and compactifications by means of monads, in: Fuzzy Logic, State of Art, Kluwer, Dortrecht/Boston/London 1993, 39-56[2] W. Gähler, Completion theory, in: Mathematisches Forschungsinstitut Oberwolfach, Tagungsbericht 48/1991, 8[3] W. Gähler, Extension structures and completions in topology and algebra, 2000
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