Galois Connection in Fuzzy Binary Relations, Applications for Discovering Association Rules and Decision Making Ali Jaoua 1 , Faisal Alvi 1 , Samir Elloumi 2 , and Sadok Ben Yahia 2 1 Department of Information and Computer Science, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia. {ajaoua, alvi}@ccse.kfupm.edu.sa 2 Facult´ e des Sciences de Tunis, D´ epartement des Sciences de l’Informatique, Campus Universitaire, Le Belv´ ed` ere, Tunis, 1060, Tunisia. {samir.elloumi, sadok.benyahia}@fst.rnu.tn Abstract. Galois connection in crisp binary relations has proved to be useful for several applications in computer science. Unfortunately, data is not always presented as a crisp binary relation but may be composed of fuzzy values, thus forming a fuzzy binary relation. This paper aims at defining the notion of fuzzy galois connection corresponding to a fuzzy binary relation in two steps: firstly by defining the term fuzzy maximal rectangle and secondly, by extending the galois lattice structure to fuzzy binary relations. Applications concerning discovery of fuzzy association rules and decision making are also presented. Keywords: Galois connection, fuzzy relation, learning, decision making, association rules. 1 Introduction Galois lattice structure and galois connection have shown their usefulness for several applications in computer science. Several papers have been published in different applied journals using the galois lattice structure of a crisp binary relation for learning, classification, information retrieval, reasoning, finding additional infor- mation, object oriented programming, database organization and automatic entity extraction [1, 2, 4]. Unfortunately, data is not always presented as a crisp binary relation but may be composed of fuzzy values, thus forming a fuzzy binary relation. In their previous attempts at fuzzy binary relation decompostion, the authors defined and extended the notion of difunctionality to fuzzy binary relations [7]. This paper aims at defining the notion of fuzzy galois connection corresponding to a fuzzy binary relation in two steps: firstly by defining the term fuzzy maximal rectangle and secondly, by extending the galois lattice structure to fuzzy binary relations. This paper is organized as follows: In Section 2, the fundamental operations and properties of fuzzy sets are recalled. In Section 3, the mathematical definitions and properties of a classical galois lattice structure are recalled. In Section 4,the notion of a fuzzy maximal rectangle is defined; then a galois lattice structure is mathematically extended to fuzzy binary relations and fuzzy galois connection is defined. In Section 5, appli- cations of fuzzy galois connection to the discovery of association rules and decision making using classification are given. 2 Mathematical Background Here we review some definitions and results that will be needed in the sequel. For details we refer to [6,7] 2.1 Fuzzy Sets Let U be a set, called the universe of discourse. An element of U is denoted by lowercase letters. A fuzzy set is defined as a collection of elements x ∈ U which includes a degree of membership for each of its elements. The membership degree for each element lies within the range [0, 1]. It may also be expressed by a membership function.