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Dedekind categories with cutoff operators. (English) Zbl 1230.18003

This paper is concerned with crispness of \(L\)-fuzzy relations. It is known that the category of \(L\)-fuzzy relations is a Dedekind category. Since there is no formula in a first-order relational language that characterizes crispness in a Dedekind category [M. Winter, Inf. Sci. 139, No. 3–4, 233–252 (2001; Zbl 0992.18004)], so, the authors introduce cutoff operators in this paper to study crispness in Dedekind categories. Cutoff operators are, intuitively, generalizations of the operator that maps a \(L\)-fuzzy relation to the largest crisp relation it contains. A representation theorem is obtained for Dedekind categories with a cutoff operator satisfying the point axiom.

MSC:

18B10 Categories of spans/cospans, relations, or partial maps
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)

Citations:

Zbl 0992.18004
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References:

[1] Bird, R.; de Moor, O., Algebra of Programming (1997), Prentice-Hall · Zbl 0867.68042
[2] Freyd, P. J.; Scedrov, A., Categories, Allegories (1990), North Holland · Zbl 0698.18002
[3] H. Furusawa, Algebraic formalisations of fuzzy relations and their representation theorems, Ph.D. Thesis, Kyushu University, 1998.; H. Furusawa, Algebraic formalisations of fuzzy relations and their representation theorems, Ph.D. Thesis, Kyushu University, 1998.
[4] Goguen, J. A., \(L\)-fuzzy sets, J. Math. Anal. Appl., 18, 145-157 (1967) · Zbl 0145.24404
[5] P.T. Johnstone, Sketches of an Elephant, A Topos Theory Compendium, vol. 1, Oxford Logic Guides, vol. 42, 2002.; P.T. Johnstone, Sketches of an Elephant, A Topos Theory Compendium, vol. 1, Oxford Logic Guides, vol. 42, 2002. · Zbl 1071.18002
[6] Jónsson, B.; Tarski, A., Boolean algebras with operators, I, II, Am. J. Math., 74, 127-162 (1952) · Zbl 0045.31601
[7] Kahl, W., Refactoring heterogeneous relation algebras around ordered categories and converse, J. Rel. Meth. Comput. Sci. (JoRMiCS), 1, 277-313 (2004)
[8] Kawahara, Y.; Furusawa, H.; Mori, M., Categorical representation theorems of fuzzy relations, Inf. Sci., 119, 3-4, 235-251 (1999) · Zbl 0943.03042
[9] Kawahara, Y.; Furusawa, H., Crispness in Dedekind categories, Bull. Inf. Cybern., 33, 1-18 (2001) · Zbl 1270.18008
[10] Olivier, J. P.; Sarrato, D., Catégories de Dedekind. Morphismes dans les categories de Schröder, C. R. Acad. Sci. Paris, 290, 939-941 (1980) · Zbl 0438.18003
[11] Popescu, A., Many-valued relation algebras, Algebra Univers., 53, 73-108 (2005) · Zbl 1086.03054
[12] Popescu, A., Some algebraic theory for many-valued relation algebras, Algebra Univers., 56, 211-235 (2007) · Zbl 1118.03061
[13] Schmidt, G.; Ströhlein, T., Relations and Graphs—Discrete Mathematics for Computer Science (1993), Springer · Zbl 0900.68328
[14] Schmidt, G.; Hattensperger, C.; Winter, M., Heterogeneous relation algebras, (Brink, C.; Kahl, W.; Schmidt, G., Relational Methods in Computer Science, Advances in Computer Science (1997), Springer: Springer Vienna), 40-54
[15] Schmidt, G.; Ströhlein, T., Relation algebras: concept of points and representability, Discrete Math., 54, 83-92 (1985) · Zbl 0575.03040
[16] Schmidt, G., Relational Mathematics (2010), Cambridge University Press
[17] Tarski, A., On the calculus of relations, J. Symbolic Logic, 6, 73-89 (1941) · JFM 67.0973.02
[18] Winter, M., A new algebraic approach to \(L\)-fuzzy relations convenient to study crispness, Inf. Sci., 139, 233-252 (2001) · Zbl 0992.18004
[19] Winter, M., Goguen categories—a categorical approach to \(L\)-fuzzy relations, (Trends in Logic, vol. 25 (2007), Springer) · Zbl 1261.03154
[20] Winter, M., Arrow categories, Fuzzy Sets Syst., 160, 2893-2909 (2009) · Zbl 1178.18002
[21] Zadeh, L. A., Fuzzy sets, Inf. Control, 8, 338-353 (1965) · Zbl 0139.24606
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