The Approaches To The Boolean Matrix And EI Algebra Representation Of Concepts

Fuzzy mathematics is based on fuzzy set theory. It is a newly arisen mathematical approach, a main mathematics tool, which is applied to study fuzzy concept. So the study of the fuzzy concept is very important. The representation of the concept is essential in the data mining. Only is the concept representation appropriate, we can mine the useful information and knowledge, and make it serve the society better. So the representation of the concept becomes the key problem of the data mining researches. The AFS theory proposed by Professor Liu Xiaodong is a new fuzzy mathematics analysis method. In which the membership function of fuzzy concept is created from the raw data of the problem, with the consistent algorithm. Using the way, AFS structure, the AFS algebraic and a negative sequence involution operation constitute the fuzzy logic system. Applying Topological molecular lattices, it depicts abstract relation between human concepts. Since fuzzy mathematics is born, definitions of the membership functions adopted by the people are all artificial and special; these functions contain too many subjective factors. But the membership function definied by AFS theory is objective and easy to be accepted, more scientific, and it is also easy to be comprehended.On the AFS theory, we define a homomorphism mapping between the EI algebra and the ring of Boolean matrix. Some attribute and some new methods of studying the Sub algebra are found out at the same time. Applying these new ways and the algebra structure of sub algebra, we can study the mathematics essence of the concepts thoroughly. On the sub-algebra of EI algebra, we have discussed the EI algebra representations of the concepts and the Boolean matrix representations of the concepts. In [24], the author had proved that for any concept on AFS structure, there is an element in EI algebra such that the concept can be represented with the EI elements, but this kind of corresponding is not unique. In this paper, we have proved that for any given concept, the EI elements corresponding to a Boolean matrix constitute a sub-algebra.The results imply that the El algebra representation is more accurate than Boolean matrix representation and we can apply these results to study more complex structure of fuzzy concept.In this paper, the AFS theory is foundation of the concept representation and homomorphism relation. So the results of this paper provided a new approach for the further study of concepts.