The multiple facets of the canonical direct implicational basis

Closure systems on a set S arises in many areas as relational databases, data-mining, formal concept analysis, artificial intelligence, logical programming or lattice theory. Implicational systems represents an efficient and convenient tool to deal with a closure system, and have been studied in various areas, with different terminology. This paper states the equality between five implicational systems issued from different works and satisfying various properties. The three main properties are the directness, canonical and minimality properties, thus the name canonical direct implicational basis given to this unique implicational system. This paper also gives the link between the canonical direct implicational basis and the Horn functions (via the prime implicants). It concludes by the necessity to compare more closely related works made independantly, and with different terminology in order to take advantage of the results really new.