The concept of Galois connection between power sets is generalized from the point of view of fuzzy logic. Studied is the case where the structure of truth values forms a complete residuated lattice. It is proved that fuzzy Galois connections are in one-to-one correspondence with binary fuzzy relations. A representation of fuzzy Galois connections by (classical) Galois connections is provided. Mathematics Subject Classification: 03B52, 04A72, 06A15, 94D05.This paper deals with Galois connections between power sets viewed from tt: perspective of fuzzy logic and fuzzy set theory. These theories develop on a formal level ZADEH'S [14] ideas of graded approach to vagueness. Up to now there are several results fulfilling the program of investigating mathematical foundations of human-like reasoning (see [3, 6, 81).A remarkable role in mathematics and in general in human reasoning is played byGalois connections (see [l, 91). We will be concerned with Galois connections between power sets of sets X and Y (shortly: Galois connection between X and Y ) , i. e., a pair ( T, 1) of mappings 1 : 2x -2' , 4 : 2* -2x , satisfying for all A , A1 , A2 E 2x,Several examples of Galois connections in mathematics can be found e.g. in [l, pp. 123 -1241. In general, a Galois connection is met whenever X is a set (of objects), Y is a set (of attributes of objects), AT (for A E ZX) is the set of all (attributes) y E Y which are related to (shared by) all (objects) 2 E A , and Bl (for B E 2' ) is the set B , B1, B2 E 2 y , ')Supported by grant no.-201/99/P060 of GA CR. 2)

We present a novel method of decomposition of an n × m binary matrix I into a Boolean product A • B of an n × k binary matrix A and a k × m binary matrix B with k as small as possible. Attempts to solve this problem are known from Boolean factor analysis where I is interpreted as an object-attribute matrix, A and B are interpreted as object-factor and factor-attribute matrices, and the aim is to find a decomposition with a small number k of factors. The method presented here is based on a theorem proved in this paper. It says that optimal decompositions, i.e. those with the least number of factors possible, are those where factors are formal concepts in the sense of formal concept analysis. Finding an optimal decomposition is an NP-hard problem. However, we present an approximation algorithm for finding optimal decompositions which is based on the insight provided by the theorem. The algorithm avoids the need to compute all formal concepts and significantly outperforms a greedy approximation algorithm for a set covering problem to which the problem of matrix decomposition is easily shown to be reducible. We present results of several experiments with various data sets including those from CIA World Factbook and UCI Machine Learning Repository. In addition, we present further geometric insight including description of transformations between the space of attributes and the space of factors.

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