The cooperative games with fuzzy coalitions in which some players act in a coalition only with a fraction of their total "power'' (endeavor, investments, material, etc.) or in which they can distribute their "power'' in more coalitions, are connected with some formal or interpretational problems. Some of these problems can be avoided if we interpret each fuzzy coalition as a fuzzy class of crisp coalitions, as shown by Mareš and Vlach in [9,10,11]. The relation between this model of fuzziness and the original one, where fuzzy coalition is a fuzzy set of players, is shown and the properties of the model are analyzed and briefly interpreted in this paper. The analysis is focused on very elementary properties of fuzzy coalitions and their payments like disjointness, superadditivity and also convexity. Three variants of their modelling are shown and their consistency is investigated. The derived results may be used for further development of the theory of fuzzy coalitions characterized by fuzzy sets of crisp coalitions. They show that the procedure developed in [11] appears to be the most adequate.
The information-theoretical entropy is an effective measure of uncertainty connected with an information source. Its transfer from the classical probabilistic information theory models to the fuzzy set theoretical environment is desirable and significant attempts were realized in the existing literature. Nevertheless, there are some open topics for analysis in the suggested models of fuzzy entropy - the main of them regard the formal aspects of the fundamental concepts. Namely their rather additive (i. e., probability-like) than monotonous (typical for fuzzy set theoretical models) structure. The main goal of this paper is to describe briefly the existing state of art, and to suggest and analyze alternative, more fuzzy set theoretical, approaches to the fuzzy entropy developed as a significant characteristic of the information sources, in the information-theoretical sense.
The development of effective methods of data processing belongs to important challenges of modern applied mathematics and theoretical information science. If the natural uncertainty of the data means their vagueness, then the theory of fuzzy quantities offers relatively strong tools for their treatment. These tools differ from the statistical methods and this difference is not only justifiable but also admissible. This relatively brief paper aims to summarize the main fuzzy approaches to vague data processing, to discuss their main advantages and also their essential limitations, and to specify their place in the wide scale of information and knowledge processing methods effective for vague data.