\({\mathcal T}\)-partitions. (English) Zbl 0930.03070

Summary: The concept of a \({\mathcal T}\)-partition is introduced as a generalization of that of a classical partition. The approach is based on the observation that for any two members of a classical semi-partition, the nonemptiness of their intersection implies their equality. This observation is generalized to \({\mathcal T}\)-semi-partitions using degrees of compatibility and equality based on a t-norm \({\mathcal T}\) and its biresidual operator \({\mathcal E}_{\mathcal T}\). By imposing an additional covering condition, the concept of a \({\mathcal T}\)-partition is obtained. An interesting numerical characterization of \({\mathcal T}\)-partitions is proved, leading to a desired one-to-one correspondence between \({\mathcal T}\)-partitions and \({\mathcal T}\)-equivalences. Moreover, the refinement of \({\mathcal T}\)-partitions is discussed. In particular, it is shown that the \({\mathcal T}^*\)-refinement of any two \({\mathcal T}\)-partitions of a given universe is again a \({\mathcal T}\)-partition of that universe if and only if the t-norm \({\mathcal T}^*\) dominates the t-norm \({\mathcal T}\).


03E72 Theory of fuzzy sets, etc.
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[1] Butnariu, D., Additive fuzzy measures and integrals, J. math. anal. appl., 93, 436-452, (1983) · Zbl 0516.28006
[2] De Baets, B.; De Cooman, G., Constructing possibility measures, (), 472-477
[3] Höhle, U., Fuzzy equalities and indistinguishability, (), 358-363
[4] Klawonn, F.; Kruse, R., From fuzzy sets to indistinguishability and back, (), A57-A59
[5] Markechová, D., The entropy of fuzzy dynamical systems and generators, Fuzzy sets and systems, 48, 351-363, (1992) · Zbl 0754.60005
[6] R. Mesiar and J. Rybárik, Entropy of fuzzy partitions, Fuzzy Sets and Systems, to appear.
[7] Piasecki, K., Probability of fuzzy events defined as denumerable additivity measure, Fuzzy sets and systems, 17, 271-284, (1985) · Zbl 0604.60005
[8] Ruspini, E., A new approach to clustering, Inform. and control, 15, 22-32, (1969) · Zbl 0192.57101
[9] Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), North-Holland New York · Zbl 0546.60010
[10] Tardiff, R., Topologies for probabilistic metric spaces, Pacific J. math., 65, 233-251, (1976) · Zbl 0337.54004
[11] Trillas, E.; Valverde, L., An inquiry into indistinguishability operators, (), 231-256 · Zbl 0564.03027
[12] Zadeh, L.A., Similarity relations and fuzzy orderings, Inform. sci., 3, 177-200, (1971) · Zbl 0218.02058
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