  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.
