On the Size of ∃-Generalized Concepts L´ eonard Kwuida 1 , Rostand S. Kuitch´ e 2 , and Romuald E. A. Temgoua 3 1 Bern University of Applied Sciences, Br¨ uckenstrasse 73, 3005 Bern, Suisse leonard.kwuida@bfh.ch 2 Universit´ e de Yaound´ e I, D´ epartement des Mathematiques, BP 812 Yaound´ e, Cameroun 3 Universit´ e de Yaound´ e I, ´ Ecole Normale Sup´ erieure, BP 47 Yaound´ e, Cameroun Abstract. Formal Concept Analysis (FCA) offers several tools for qual- itative data analysis. One possibility is to group objects that share com- mon attributes together and get a concept lattice that describes the data. Quite often the size of this concept lattice is very large. Many authors have investigated methods to reduce the size of this lattice. In [KMBV14] the authors consider putting together some attributes to reduce the size of the attribute sets. But this reduction does not always carry over the set of concepts. They have provided some counter examples where the size of the concept lattice increases by one after putting two attributes together. Then they asked the following question: ”How many new concepts can be generated by an ∃-generalization on just two attributes?” The present paper provides a family of contexts for which the size increases on more than one concept after putting solely two attributes together. Keywords : Formal Concept Analysis; Concept Lattices; Generalizing Attributes; 1 Introduction An elementary information system can be represented by a set G of objects or entities, a set M of attributes or characteristics together with an incidence relation I that encodes whether an object g ∈ G has an attribute m ∈ M . For such a system we write (g,m) ∈ I or g I m to mean that the object g has the attribute m. The binary relation K := (G,M, I) is called a formal context, and describes an elementary information system. To extract knowledge from such information systems, one possibility is to get clusters of objects and/or attributes by grouping together those sharing the same characteristics. These pairs, called concepts, were formalized by Rudolf Wille [Wi82]. Traditional philosophers consider a concept as defined by two parts: an extent and an intent. The extent contains all entities belonging to the concept and the intent is the set of all attributes common to all entities in the concept. To formalize the notion of concept the following notations has been arXiv:1709.08060v1 [math.LO] 23 Sep 2017