Border algorithms for computing Hasse diagrams of arbitrary lattices

Abstract

Lattices are mathematical structures with many applications in computer science; among these, we are interested in fields like data mining, machine learning, or knowledge discovery in databases. One well-established use of lattice theory is in formal concept analysis (FCA), where the concept lattice with its diagram graph allows the visualization and summarization of data in a more concise representation. In the Data Mining community, the same mathematical notions (often under additional “frequency” constraints that bound from below the size of the support set) are studied under the banner of Closed-Set Mining. In these applications, each dataset consists of transactions, also called objects, each of which, besides having received a unique identifier, consists of a set of items or attributes taken from a previously agreed finite set. A concept is a pair formed by a set of transactions —the extent set or support set of the concept— and a set of attributes —the intent set of the concept— defined as the set of all those attributes that are shared by all the transactions present in the extent. Some data analysis processes are based on the family of all intents (the “closures” stemming from the dataset); but others require to determine also their order relation, which is a finite lattice, in the form of a line graph (the Hasse diagram).