Chapter 16 Fundamental Mathematical Notions of the Theory of Socially Embedded Games: A Granular Computing Perspective
Anna Oomolinska · 2004
Summary. The aim of this article is to present the key mathematical notions of the theory of socially embedded (or generalized) games (GGT) such as rule, rule application, and rule complex from the standpoint of granularity of information. In GGT, social actors as well as social games and, more generally, interactions, are represented by granules of information called rule complexes. Rule complexes are special cases of complexes of points of some space where the points are rules over a considered language. In many cases, rule complexes may be viewed as granules of information just because the rules constituting the rule com plexes are drawn together on the base of similarity and/or functionality. On the other hand, rules represent granules of information. Therefore, rule complexes are multilevel structures representing granules of information as well. In this chapter, we also discuss mereological questions, and, in particular, we define three kinds of crisp ingredient-whole relationships with respect to complexes of points (viz., being of a g-element, being of a g-subset, and being of a subcomplex of a complex of points) and present some ideas on how to define the notion of a part (ingredient) to a degree in cases of complexes of points.