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Keywords:
fuzzy set over $MV$-lagebra; complete subobjects; subobjects classification
fuzzy set over $MV$-lagebra; complete subobjects; subobjects classification
Summary:
A subobjects structure of the category $\Omega $- of $\Omega $-fuzzy sets over a complete $MV$-algebra $\Omega =(L,\wedge ,\vee ,\otimes ,\rightarrow )$ is investigated, where an $\Omega $-fuzzy set is a pair ${\mathbf A}=(A,\delta )$ such that $A$ is a set and $\delta \:A\times A\rightarrow \Omega $ is a special map. Special subobjects (called complete) of an $\Omega $-fuzzy set ${\mathbf A}$ which can be identified with some characteristic morphisms ${\mathbf A}\rightarrow \Omega ^*=(L\times L,\mu )$ are then investigated. It is proved that some truth-valued morphisms $\lnot _{\Omega }\:\Omega ^*\rightarrow \Omega ^*,\cap _{\Omega }$, $\cup _{\Omega } \:\Omega ^*\times \Omega ^*\rightarrow \Omega ^*$ are characteristic morphisms of complete subobjects.
References:
[1] M. Eytan: Fuzzy sets: a topos-logical point of view. Fuzzy Sets and Systems 5 (1981), 47–67. DOI 10.1016/0165-0114(81)90033-6 | MR 0595953 | Zbl 0453.03059
[2] J. A. Goguen: L-fuzzy sets. J. Math. Anal. Appl. 18 (1967), 145–174. DOI 10.1016/0022-247X(67)90189-8 | MR 0224391 | Zbl 0145.24404
[3] R. Goldblatt: TOPOI, The Categorical Analysis of Logic. North-Holland Publ. Co., Amsterdam-New York-Oxford, 1979. MR 0551362
[4] D. Higgs: A Category Approach to Boolean-Valued Set Theory. Manuscript, University of Waterloo, 1973.
[5] 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. MR 1345643
[6] 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. MR 1154568
[7] U. Höhle: Classification of subsheaves over GL-algebras. Proceedings of Logic Colloquium 98, Prague, Springer Verlag, 1999. MR 1743263
[8] 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. MR 1345641
[9] U. Höhle: Monoidal closed categories, weak topoi and generalized logics. Fuzzy Sets and Systems 42 (1991), 15–35. DOI 10.1016/0165-0114(91)90086-6 | MR 1123574
[10] P. T. Johnstone: Topos Theory. Academic Press, London-New York-San Francisco, 1977. MR 0470019 | Zbl 0368.18001
[11] Toposes, Algebraic Geometry and Logic. F. W. Lawvere (ed.), Springer-Verlag, Berlin-Heidelberg-New York, 1971. MR 0330254
[12] M. Makkai and E. G. Reyes: Firts Order Categorical Logic. Springer-Verlag, Berlin-New York-Heidelberg, 1977. MR 0505486
[13] A. M. Pitts: Fuzzy sets do not form a topos. Fuzzy Sets and Systems 8 (1982), 101–104. DOI 10.1016/0165-0114(82)90034-3 | MR 0665492 | Zbl 0499.03051
[14] D. Ponasse: Categorical studies of fuzzy sets. Fuzzy Sets and Systems 28 (1988), 235–244. DOI 10.1016/0165-0114(88)90031-0 | MR 0976664 | Zbl 0675.03032
[15] A. Pultr: Fuzzy mappings and fuzzy sets. Comment. Mat. Univ. Carolin. 17 (1976), . MR 0416923 | Zbl 0343.02048
[16] A. Pultr: Closed Categories of L-fuzzy Sets. Vortrage zur Automaten und Algorithmentheorie, TU Dresden, 1976.
[17] L. N. Stout: A survey of fuzzy set and topos theory. Fuzzy Sets and Systems 42 (1991), 3–14. DOI 10.1016/0165-0114(91)90085-5 | MR 1123573 | Zbl 0738.04002
[18] 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. MR 1345646 | Zbl 0827.03039
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