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

De Baets, Bernard; Mesiar, Radko

Fuzzy Sets Syst. 97, No. 2, 211-223 (1998).

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 \({\mathcal T}\).

MSC:

03E72

Theory of fuzzy sets, etc.

Keywords:

degree of compatibility; degree of equality; biresidual operator; dominating t-norm; residual implicator; \({\mathcal T}\)-partition; \({\mathcal T}\)-semi-partitions; covering; \({\mathcal T}\)-equivalences

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