On Finity, Countability, Cardinalities, and Cylindric Extensions of Type-2 Fuzzy Sets in Linguistic Summarization of Databases

The discussion in this paper is closely related to our idea of a type-2 linguistic summary of a database presented by Niewiadomski [A. Niewiadomski, ??A type-2 fuzzy approach to linguistic summarization of data,?? IEEE Trans. Fuzzy Syst., vol. 16, no. 1, pp. 198-213, Feb. 2008], which is intended to be an efficient tool of knowledge discovery from large databases. In that approach, we put emphasis on algorithms and applications of type-2 fuzzy sets in summarizing databases. Hence, we have implicitly assumed and considered linguistic expressions represented by sets in discrete (or even finite) universes of discourse. Now, in this paper, we discuss more thoroughly some of the properties and formal aspects of both discrete and continuous type-2 fuzzy sets that represent the elements of linguistic summaries, i.e., quantifiers, summarizers, and/or qualifiers. We underline differences between type-2 fuzzy sets in finite and in infinite (usually continuous) universes of discourse and, henceforth, between their cardinalities and similar measures, which are used in evaluating the goodness (quality) of type-2 linguistic summaries. In addition, we define new imprecision measures of linguistic expressions represented by type-2 fuzzy sets, and propose a definition of the cylindric extension of a type-2 fuzzy set. Finally, we apply these new concepts to generalize algorithms of type-2 linguistic summarization and propose some new features in comparison to our previous approach [A. Niewiadomski, ??A type-2 fuzzy approach to linguistic summarization of data,?? IEEE Trans. Fuzzy Syst., vol. 16, no. 1, pp. 198-213, Feb. 2008].