Fuzzy Sets and Algebraic Hyperoperations to Model Interpersonal Relations
In Social Sciences propositions and descriptions of interpersonal relationships are given by means of linguistic expressions, which cannot be formalized with the classic binary logic. Then a necessary tool are fuzzy sets that give the possibility to measu
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tract In Social Sciences propositions and descriptions of interpersonal relationships are given by means of linguistic expressions, which cannot be formalized with the classic binary logic. Then a necessary tool are fuzzy sets that give the possibility to measure the degree of belonging of an element to a set described by a linguistic property or the degree of a relation between individuals. Moreover many times there is uncertainty on the result of an aggregation operation and we can have the necessity to consider together many possible results of the interaction of any ordered pair of elements, e.g. individuals. We propose the algebraic hyperoperations as very useful instruments to manage these types of uncertainty. We show as fuzzy relations and hypergroupoids permit to have an efficient representations of many aspects of social phenomena. Keywords Social relations models for social sciences
Fuzzy relations Hypergroupoids Mathematical
1 Introduction Social Relations are described with mathematical models by many authors (e.g. Moreno 1934, 1951). A more complete description can be obtained if we utilize instruments to manage semantic uncertainty. At this aim we propose to introduce models based on fuzzy sets and algebraic hyperoperations. The first one give the possibility to measure the degree of belonging of an element to a set described by a Š. Hošková-Mayerová (&) Department of Mathematics and Physics, University of Defence Brno, Kounicova 65, 66210 Brno, Czech Republic e-mail: [email protected] A. Maturo Department of Humanities, Arts and Social Sciences, University of Pescara-Chieti, Via dei Vestini 31, 66100 Chieti, Italy e-mail: [email protected] © Springer International Publishing Switzerland 2017 A. Maturo et al. (eds.), Recent Trends in Social Systems: Quantitative Theories and Quantitative Models, Studies in Systems, Decision and Control 66, DOI 10.1007/978-3-319-40585-8_19
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Š. Hošková-Mayerová and A. Maturo
linguistic property or the degree of a relation between individuals, the second one permit to consider together many possible results of the interaction of any ordered pair of elements. In Social Sciences these aspects are very important and the instrument we propose can be very efficient for modelling the complexity of social phenomena. The paper is organized as follows: Firstly, in Sect. 2, we review some basic definitions on fuzzy sets and fuzzy relations and discuss the modelling of social relations with fuzzy sets. In Sect. 3, we give an introduction to the theory of algebraic hyperstructures, its history and motivation, and discuss the formalizing social relations with this mathematical tool. In particular we introduce some particular social relations and the correspondent associated algebraic hyperstructures for modelling these relations. In Sect. 4, we study some mathematical properties of the introduced relations and hypergroupoids and their social meanings. Finally, in Sect. 5 we present some conclusions and perspective of research in modelling problems of social behavi
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