| Title:
|
Complete subobjects of fuzzy sets over $MV$-algebras (English) |
| Author:
|
Močkoř, Jiří |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
54 |
| Issue:
|
2 |
| Year:
|
2004 |
| Pages:
|
379-392 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
fuzzy set over $MV$-lagebra |
| Keyword:
|
complete subobjects |
| Keyword:
|
subobjects classification |
| MSC:
|
03E72 |
| MSC:
|
06D15 |
| MSC:
|
18B05 |
| idZBL:
|
Zbl 1080.18001 |
| idMR:
|
MR2059258 |
| . |
| Date available:
|
2009-09-24T11:13:28Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127895 |
| . |
| Reference:
|
[1] M. Eytan: Fuzzy sets: a topos-logical point of view.Fuzzy Sets and Systems 5 (1981), 47–67. Zbl 0453.03059, MR 0595953, 10.1016/0165-0114(81)90033-6 |
| Reference:
|
[2] J. A. Goguen: L-fuzzy sets.J. Math. Anal. Appl. 18 (1967), 145–174. Zbl 0145.24404, MR 0224391, 10.1016/0022-247X(67)90189-8 |
| Reference:
|
[3] R. Goldblatt: TOPOI, The Categorical Analysis of Logic.North-Holland Publ. Co., Amsterdam-New York-Oxford, 1979. MR 0551362 |
| Reference:
|
[4] D. Higgs: A Category Approach to Boolean-Valued Set Theory.Manuscript, University of Waterloo, 1973. |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[7] U. Höhle: Classification of subsheaves over GL-algebras.Proceedings of Logic Colloquium 98, Prague, Springer Verlag, 1999. MR 1743263 |
| Reference:
|
[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 |
| Reference:
|
[9] U. Höhle: Monoidal closed categories, weak topoi and generalized logics.Fuzzy Sets and Systems 42 (1991), 15–35. MR 1123574, 10.1016/0165-0114(91)90086-6 |
| Reference:
|
[10] P. T. Johnstone: Topos Theory.Academic Press, London-New York-San Francisco, 1977. Zbl 0368.18001, MR 0470019 |
| Reference:
|
[11] : Toposes, Algebraic Geometry and Logic.F. W. Lawvere (ed.), Springer-Verlag, Berlin-Heidelberg-New York, 1971. MR 0330254 |
| Reference:
|
[12] M. Makkai and E. G. Reyes: Firts Order Categorical Logic.Springer-Verlag, Berlin-New York-Heidelberg, 1977. MR 0505486 |
| Reference:
|
[13] A. M. Pitts: Fuzzy sets do not form a topos.Fuzzy Sets and Systems 8 (1982), 101–104. Zbl 0499.03051, MR 0665492, 10.1016/0165-0114(82)90034-3 |
| Reference:
|
[14] D. Ponasse: Categorical studies of fuzzy sets.Fuzzy Sets and Systems 28 (1988), 235–244. Zbl 0675.03032, MR 0976664, 10.1016/0165-0114(88)90031-0 |
| Reference:
|
[15] A. Pultr: Fuzzy mappings and fuzzy sets.Comment. Mat. Univ. Carolin. 17 (1976), . Zbl 0343.02048, MR 0416923 |
| Reference:
|
[16] A. Pultr: Closed Categories of L-fuzzy Sets.Vortrage zur Automaten und Algorithmentheorie, TU Dresden, 1976. |
| Reference:
|
[17] L. N. Stout: A survey of fuzzy set and topos theory.Fuzzy Sets and Systems 42 (1991), 3–14. Zbl 0738.04002, MR 1123573, 10.1016/0165-0114(91)90085-5 |
| Reference:
|
[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. Zbl 0827.03039, MR 1345646 |
| . |
