Social bookmarking tools are rapidly emerging on the Web. In such systems users are setting up lightweight conceptual structures called folksonomies. Unlike ontologies, shared conceptualizations are not formalized, but rather implicit. We present a new data mining task, the mining of all frequent tri-concepts, together with an efficient algorithm, for discovering these implicit shared conceptualizations. Our approach extends the data mining task of discovering all closed itemsets to three-dimensional data structures to allow for mining folksonomies. We provide a formal definition of the problem, and present an efficient algorithm for its solution. Finally, we show the applicability of our approach on three large real-world examples.

Formal Concept Analysis, developed during the last fifteen years, has been based on the dyadic understanding of a concept constituted by its extension and its intension. The pragmatic philosophy of Charles S. Peirce with his three universal categories, and experiences in data analysis, have suggested a triadic approach to Formal Concept Analysis. This approach starts with the primitive notion of a triadic context defined as a quadruple (G, M, B, Y) where G, M, and B are sets and Y is a ternary relation between G, M, and B, i.e. Y G×M×B; the elements of G, M, and B are called objects, attributes, and conditions, respectively, and (g, m,b) Y is read: the object g has the attribute m under (or according to) the condition b. A triadic concept of a triadic context (G, M, B, Y) is defined as a triple (A