Abstract. In this paper, the ordered set of rough sets determined by a quasiorder relation R is investigated. We prove that this ordered set is a complete, completely distributive lattice. We show that on this lattice can be defined three different kinds of complementation operations, and we describe its completely join-irreducible elements. We also characterize the case in which this lattice is a Stone lattice. Our results generalize some results of J. Pomyka la and J. A. Pomyka la (1988) and M. Gehrke and E. Walker (1992) in case R is an equivalence.

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Abstract. In this paper we show that, during an elementary extension of a context, each of the classification trees of the newly created box extent lattice can be obtained by modifying the classification trees of the box extent lattice of the original, smaller context. We also devise an algorithm which, starting from a classification tree of the box extent lattice of the smaller context (H, M, I ∩ H × M ), gives a classification tree of the extended context (G, M, I) which contains the new elements inserted. The efficiency of the method is given by the fact that it is sufficient to know the original context, the classification tree of the box extent lattice and its box extents while the knowledge of a new box extension of the extended context mesh elements is not required (except for one, which is the new element box extension). (G, M, I) where G and M are sets and I ⊆ G × M is a binary relation. The elements of G and M are called objects and attributes of the context, respectively. The relation gIm means that the object g has the attribute m. A small context can be easily represented by a cross table, i.e., by a rectangular table with rows headed by the object names and the columns by the attribute names. A cross in the intersection of row g and column m, means that object g has attribute m. For all sets A ⊆ G and B ⊆ M , we define A = {m ∈ M | g I m for all g ∈ A}, B = {g ∈ G | g I m for all m ∈ B}. Preliminaries: Box lattice, extent lattice A context ([2]) is a triple

This paper presents a mathematical approach which can be successfully applied in different group technology problems. The method is originated from formal concept analysis (FCA). We produce some particular extent partitions of a formal context which encodes the relationship between given technical objects and their attributes. We show how to define the part type notions and also how to solve the cell formation problem by means of these extent partitions. The main focus of the paper is a solution method for the determination of master parts. The method is based on notions defined by modal operators analogously to formal concepts and uses the extent partitions of the complementary context. It follows from the nature of the problem that several solutions can exist; a consistency measure is introduced in order to rank them. [

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