Incremental construction of Alpha lattices and association rules Henry Soldano 1 , V´ eronique Ventos 2 , Marc Champesme 1 , David Forge 2 1 L.I.P.N, UMR-CNRS 7030, Universit´ e Paris-Nord, 93430 Villetaneuse, France 2 LRI, UMR-CNRS 8623, Universit´ e Paris-Sud, 91405 Orsay, France Abstract. In this paper we discuss Alpha Galois lattices (Alpha lattices for short) and the corresponding association rules. An alpha lattice is coarser than the related concept lattice and so contains fewer nodes, so fewer closed patterns, and a smaller basis of association rules. Coarseness depends on a a priori classification, i.e. a cover C of the powerset of the instance set I , and on a granularity parameter α. In this paper, we define and experiment a Merge operator that when applied to two Alpha lattices G(C1,α) and G(C2,α) generates the Alpha lattice G(C1 ∪C2,α), so leading to a class-incremental construction of Alpha lattices. We then briefly discuss the implementation of the incremental process and describe the min-max bases of association rules extracted from Alpha lattices. 1 Introduction Galois lattices (or concept lattices ) explicitly represent all subsets of a set I of instances that are distinguishable according to a given language L, whose elements will be called terms or patterns. A node of a concept lattice is composed of a set of instances (the extent) and a term (the intent). The intent is the most specific term which recognizes the instances of the extent, and the extent is the greatest set of instances so recognized. This means that a fraction of the language (the set of intents) represents all the distinguishable subsets of instances. Such an intent is also denoted as a closed term, and as a frequent closed term when only closed terms, whose extents are large enough according to a bound minsupp ∗|I |, are considered[8]. In Data Mining the problem of finding frequent itemsets, an essential part of the process of extracting association rules, is therefore reduced to finding closed frequent itemsets. Many research efforts are currently spent in finding efficient algorithms for that purpose[12]. More recently, in order to deal with semi-structured data, more expressive patterns than itemsets have been heavily investigated [7,5]. Again closure operators have been used to efficiently extract closed patterns, as for instance attribute trees [1, 11]. Still the number of closed patterns is often too high when dealing with real- world data, and some way to select them has to be found [6]. Recently, Alpha Galois lattices (Alpha lattices for short) have been defined [16]. Each node of an Alpha lattice corresponds to an Alpha closed term i.e. a closed term that satisfies a local frequency constraint, therefore allowing a better control on the selection