| Title:
|
A characterization of tribes with respect to the Łukasiewicz $t$-norm (English) |
| Author:
|
Klement, Erich Peter |
| Author:
|
Navara, Mirko |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
47 |
| Issue:
|
4 |
| Year:
|
1997 |
| Pages:
|
689-700 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We give a complete characterization of tribes with respect to the Łukasiewicz $t$-norm, i. e., of systems of fuzzy sets which are closed with respect to the complement of fuzzy sets and with respect to countably many applications of the Łukasiewicz $t$-norm. We also characterize all operations with respect to which all such tribes are closed. This generalizes the characterizations obtained so far for other fundamental $t$-norms, e. g., for the product $t$-norm. (English) |
| MSC:
|
04A72 |
| MSC:
|
28A20 |
| MSC:
|
28E10 |
| MSC:
|
46S10 |
| MSC:
|
46S99 |
| idZBL:
|
Zbl 0902.28015 |
| idMR:
|
MR1479313 |
| . |
| Date available:
|
2009-09-24T10:09:35Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127387 |
| . |
| Reference:
|
[BK] Butnariu, D., Klement, E.P.: Triangular norm-based measures and games with fuzzy coalitions.Kluwer, Dordrecht, 1993. |
| Reference:
|
[F] Frank, M.J.: On the simultaneous associativity of $F(x,y)$ and $x+y-F(x,y)$.Aequationes Math. 19 (1979), 194–226. Zbl 0444.39003, MR 0556722, 10.1007/BF02189866 |
| Reference:
|
[K] Klement, E.P.: Construction of fuzzy $\sigma $-algebras using triangular norms.J. Math. Anal Appl. 85 (1982), 543–565. Zbl 0491.28003, MR 0649189, 10.1016/0022-247X(82)90015-4 |
| Reference:
|
[M:semi] Mesiar, R.: Fundamental triangular norm based tribes and measures.J. Math. Anal. Appl. 177 (1993), 633–640. Zbl 0816.28014, MR 1231507, 10.1006/jmaa.1993.1283 |
| Reference:
|
[M] Mesiar, R.: On the structure of $T_s$-tribes.Tatra Mountains Math. Publ. 3 (1993), 167–172. MR 1278531 |
| Reference:
|
[MN] Mesiar, R., Navara, M.: $T_s$-tribes and $T_s$-measures.J. Math. Anal. Appl (to appear). MR 1397888 |
| Reference:
|
[N] Navara, M.: A characterization of triangular norm based tribes.Tatra Mountains Math Publ. 3 (1993), 161–166. Zbl 0799.28013, MR 1278530 |
| Reference:
|
[Pykacz] Pykacz, J.: Fuzzy set ideas in quantum logics.Int. J. Theor. Physics 31 (1992), 1765–1781. Zbl 0789.03049, MR 1183522, 10.1007/BF00671785 |
| Reference:
|
[SS] Schweizer, B., Sklar, A.: Probabilistic Metric Spaces.North-Holland, New York, 1983. MR 0790314 |
| . |
