The joy of implications, aka pure Horn functions: mainly a survey

Abstract

Apart from a brief look at applications (Relational Databases, Formal Concept Analysis, data mining) this article is devoted to the mathematical t h e o r y of implications (=pure Horn formulas). It is mainly a survey of results obtained in the last thirty years, but features a few novelties as well. Some keywords: The Duquenne-Guiges (implicational) base, the canonical direct base, prime implicates, the consensus method, implications and meet irreducible closed sets, optimum bases for certain lattices, ordered direct bases, generating all closed sets, general (i.e. impure) Horn functions. We pose four open problems to stimulate further research.