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Quotients with respect to similarity relations. (English) Zbl 0666.18002
The problem of the quotient w.r.t. a similarity relation is closely related to Poincaré’s paradox and central problems in cluster analysis. Amazingly this problem remains open for more than 30 years. As a generalization of strong M-categories we introduce the concept of M- valued sets, which leads to the definition of the category M-SET. In M- SET we give a complete solution of the quotient problem.

MSC:
18B10 Categories of spans/cospans, relations, or partial maps
06F05 Ordered semigroups and monoids
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
62H30 Classification and discrimination; cluster analysis (statistical aspects)
94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
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References:
[1] Behrens, E.-A, Ring theory, (1972), Academic Press New York
[2] Birkhoff, G, Lattice theory, () · Zbl 0126.03801
[3] Dubuc, E.J, Kan extensions in enriched category theory, () · Zbl 0228.18002
[4] Duran, B.S; Odell, P.L, Cluster analysis, () · Zbl 0256.62053
[5] Eilenberg, S; Kelly, G.M, Closed categories, () · Zbl 0192.10604
[6] Fourman, M.P; Scott, D.S, Sheaves and logic, (), 302-401 · Zbl 0415.03053
[7] Frink, O, New algebras of logic, Amer. math. monthly, 45, 210-219, (1938) · JFM 64.0030.03
[8] Fuchs, L, Über die ideale arithmetischer ringe, Commentarii math. helv., 23, 334-341, (1949) · Zbl 0040.30103
[9] Higgs, D, A category approach to Boolean-valued set theory, (1973), Preprint, Waterloo
[10] Höhle, U, The category CM-SET, an algebraic approach to uncertainty, (1986), Preprint, Cincinnati
[11] Janowitz, M.S, An order theoretic model for cluster analysis, SIAM J. appl. math., 34, 55-72, (1978) · Zbl 0379.62050
[12] Hardine, N; Sibson, R, Mathematical taxonomy, (1977), Wiley New York
[13] Johnstone, P.T, Stone spaces, (1982), Cambridge University Press London · Zbl 0499.54001
[14] Lawvere, F.W, Metric spaces, generalized logic and closed categories rend., Sem. mat. fisico milano, 43, 135-166, (1974) · Zbl 0335.18006
[15] Lowen, R, On the relation between fuzzy equalities, pseudometrics and uniformities, Quaestiones math., 7, 407-419, (1984) · Zbl 0567.54004
[16] Menger, K, Probabilistic geometry, (), 226-229 · Zbl 0042.37201
[17] Menger, K, Probabilistic theories of relations, (), 178-180 · Zbl 0042.37103
[18] Menger, K, Geometry and positivism, a probabilistic microgeometry, (), 225-234
[19] Negoita, C.V; Ralescu, D.A, Representation theorems for fuzzy concepts, Kybernetics, 4, (1975) · Zbl 0352.02044
[20] Poincaré, H, La science et l’hypothèse, (1902), Flammarion Paris · JFM 34.0080.12
[21] Poincaré, H, La valeur de la science, (1904), Flammarion Paris · JFM 33.0071.03
[22] Ralescu, D.A, Fuzzy subobjects in a category and the theory of C-sets, Fuzzy sets and systems, 1, 193-202, (1978) · Zbl 0382.18003
[23] Schweizer, B; Sklar, A, Probabilistic metric spaces, (1983), North-Holland Amsterdam · Zbl 0546.60010
[24] Scott, D.S, A proof of the independence of the continuum hypothesis, Math. systems theory, 1, 89-111, (1967) · Zbl 0149.25302
[25] Valverde, L, On the structure of F-indistinguishability operators, Fuzzy sets and systems, 17, 313-328, (1985) · Zbl 0609.04002
[26] Zadeh, L.A, Similarity relations and fuzzy orderings, Inform. sci., 3, 177-200, (1971) · Zbl 0218.02058
[27] Zarsiki, O; Samuel, P, Commutative algebra, (1958), Van Nostrand New York
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