The Operations Of Strong Direct Product And Addition Of Formal Contexts

Formal context is one of the core notions in the theory of Formal Concept Analysis. Thereare close relationships between the new formal contexts, which are generated from the opera-tions of formal contexts, and these original contexts.This paper constructs an operation called the strong direct product of n formal contexts,and then discusses the relationships between the product context of n contexts ( abbreviated ton-SDPFC ) and its n original contexts from four aspects: concept lattice, implications betweenattributes, irreducible elements and attribute characteristics, rough sets in context. It is provedthat the projection of every formal concept of a n-SDPFC on the i-th original formal context isalso a concept of the corresponding original formal context, and under some given conditions,the Cartesian product of n formal concepts of the n corresponding original formal contextsis a concept of the n-SDPFC, too. An implication between attributes of an n-SDPFC can beobtained if we know all of the implications between attributes in the original formal contexts,since it can be formed component-wisely; and vice verse. A judgement theorem for irreducibleelements in the concept lattice of an n-SDPFC is obtained, which is based on the irreducibleelements in the n original concept lattices. Consequently, a method for the determination ofthe attribute characteristics of the n-SDPFC is gained. In some given circumstances, the upperand lower approximations of sets in an n-SDPFC can be attained component-wisely. Basedon all studied above, the main characteristic of the n-SDPFC is investigated, then an methodto decompose the contexts with such characteristic is proposed, which decreases our workloadenormously.Addition is another operation of formal contexts. Here the subject may be dealt with fromtwo aspects: the apposition and the subposition. Actually, the apposition is the combination offormal contexts in a vertical way and the subposition is the combination of formal contexts in ahorizontal way. The combined formal context has close relationships with the original contextsin the following respects: attribute characteristics, rule acquisition, irreducible attributes, and soon. These relationships are given in details in terms of formal context without or with decision.