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Special properties, closures and interiors of crisp and fuzzy relations. (English) Zbl 0664.04001

This paper offers optimal definitions of the special properties which relations on a set may possess, and which combine to define the classically important types of preorders, both kinds of orders, tolerances and equivalences. The useful concept of local properties is introduced. The concepts of closure and interior of a relation with respect to a property are defined. Finally, fast algorithms are given for the computer calculation of the more troublesome closures, and a standardized procedure for the analysis of local orders is presented.
Reviewer: Li Sang Ho

MSC:

03E20 Other classical set theory (including functions, relations, and set algebra)
03-04 Software, source code, etc. for problems pertaining to mathematical logic and foundations
03E72 Theory of fuzzy sets, etc.
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[1] Bandler, W.; Kohout, L.J., The use of new relational products in clinical modelling, (), 240-246
[2] Bandler, W.; Kohout, L.J., Fuzzy power sets and fuzzy implication operators, Fuzzy sets and systems, 4, 13-30, (1980) · Zbl 0433.03013
[3] Bandler, W.; Kohout, L.J., The identification of hierarchies in symptoms and patients through computation of fuzzy relational products, (), 191-194
[4] Bandler, W.; Kohout, L.J., A survey of fuzzy relational products in their applicability to medicine and clinial psychology, (), 107-118
[5] Borůvka, O., Grundlagen der gruppoid- und gruppentheorie, (1960), VEB Deutscher Verlag der Wissenschaften Berlin, Translated as: Groupoids and Groups (VEB Deutscher Verlag der Wissenschaften, Berlin, 1974) · Zbl 0091.02001
[6] Kallala, M.; Kohout, L.J., A two-stage method for automatic handwriting classification by means of norms and fuzzy relational inference, (), 312-323
[7] Kaufmann, A.; Kaufmann, A., (), Translated as:
[8] Schröder, E., Vorlesungen über die algebra der logik, III. Band, (1966), (Leipzig, 1895). Republished Chelsea, New York
[9] Zadeh, L.A., Similarity relations and fuzzy orderings, Inform. sci., 3, 177-200, (1971) · Zbl 0218.02058
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