Abstract
A subobjects structure of the category Ω-FSet of Ω-fuzzy sets over a complete MV-algebra \(\Omega = \left( {L,\Lambda , \vee , \otimes , \to } \right)\) is investigated, where an Ω-fuzzy set is a pair A = (A, δ) such that A is a set and δ: A × A → Ω is a special map. Special subobjects (called complete) of an Ω-fuzzy set A which can be identified with some characteristic morphisms A → Ω* = (L × L, μ) are then investigated. It is proved that some truth-valued morphisms \(_\Omega :\Omega ^ * \to \Omega ^ * , \cap _\Omega , \cup _\Omega :\Omega ^ * \times \Omega ^ * \to \Omega ^ * \) are characteristic morphisms of complete subobjects.
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References
M. Eytan: Fuzzy sets: a topos-logical point of view. Fuzzy Sets and Systems 5 (1981), 47-67.
J. A. Goguen: L-fuzzy sets. J. Math. Anal. Appl. 18 (1967), 145-174.
R. Goldblatt: TOPOI, The Categorical Analysis of Logic. North-Holland Publ. Co., Amsterdam-New York-Oxford, 1979.
D. Higgs: A Category Approach to Boolean-Valued Set Theory. Manuscript, University of Waterloo, 1973.
U. Höhle: Presheaves over GL-monoids. In: Non-Classical Logic and Their Applications to Fuzzy Subsets. Kluwer Academic Publ., Dordrecht-New York, 1995, pp. 127-157.
U. Höhle: M-Valued sets and sheaves over integral, commutative cl-monoids. Applications of Category Theory to Fuzzy Subsets. Kluwer Academic Publ., Dordrecht-Boston, 1992, pp. 34-72.
U. Höhle: Classification of subsheaves over GL-algebras. Proceedings of Logic Colloquium 98, Prague. Springer Verlag, 1999.
U. Höhle: Commutative, residuated l-monoids. Non-Classical Logic and Their Applications to Fuzzy Subsets. Kluwer Academic Publ., Dordrecht-New York, 1995, pp. 53-106.
U. Höhle: Monoidal closed categories, weak topoi and generalized logics. Fuzzy Sets and Systems 42 (1991), 15-35.
P. T. Johnstone: Topos Theory. Academic Press, London-New York-San Francisco, 1977.
Toposes, Algebraic Geometry and Logic (F. W. Lawvere, ed.). Springer-Verlag, Berlin-Heidelberg-New York, 1971.
M. Makkai and E. G. Reyes: Firts Order Categorical Logic. Springer-Verlag, Berlin-NewYork-Heidelberg, 1977.
A.M. Pitts: Fuzzy sets do not form a topos. Fuzzy Sets and Systems 8 (1982), 101-104.
D. Ponasse: Categorical studies of fuzzy sets. Fuzzy Sets and Systems 28 (1988), 235-244.
A. Pultr: Fuzzy mappings and fuzzy sets. Comment. Mat. Univ. Carolin. 17 (1976).
A. Pultr: Closed Categories of L-fuzzy Sets. Vortrage zur Automaten und Algorithmen-theorie, TU Dresden, 1976.
L. N. Stout: A survey of fuzzy set and topos theory. Fuzzy Sets and Systems 42 (1991), 314.
O. Wyler: Fuzzy logic and categories of fuzzy sets. Non-Classical Logic and Their Applications to Fuzzy Subsets. Kluwer Academic Publ., Dordrecht-New York, 1995, pp. 235-268.
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Močkoř, J. Complete Subobjects of Fuzzy Sets Over MV-Algebras. Czechoslovak Mathematical Journal 54, 379–392 (2004). https://doi.org/10.1023/B:CMAJ.0000042376.21044.1a
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DOI: https://doi.org/10.1023/B:CMAJ.0000042376.21044.1a