The base of finite EI algebra

Abstract

In many fuzzy systems, the membership functions of fuzzy concepts are obtained by the expert's experiences. Different persons may have different membership functions for the same fuzzy concept. And the current algorithms for determining membership functions are not universal. AFS (axiomatic fuzzy sets) theory, which is a new approach, can improve this defection. AFS theory can be used to study the natural language and the law of human thinking and also can be easily operated by computers. In AFS theory, the mathematical structures of fuzzy concepts have been studied by the EI algebra, a molecular lattice defined by Wang Guojun (1992), and AFS structures which are super-graph. It is proved that any fuzzy concept on the finite universe of discourse is the EI representations of some simple concepts. In this paper, the authors define the base of the EI algebra and EI independence and apply them to study the algebraic structures of the EI algebra. The authors also give some propositions about the base for some EI algebra. All these can be applied in structure analysis of concepts. The base of sub algebra of EI algebra can simplify the EI algebra representations of human concepts and improve the intelligent capability of computer. At the same time, it has potential applications in fuzzy clustering analysis and data mining. At last the authors demonstrate how to use the elements of EI algebra to study the natural language by a real world example.