About the family of closure systems preserving non-unit implications in the Guigues-Duquenne base

Abstract

Consider a Guigues-Duquenne base ∑F = ∑JF ∪ ∑ ↓F of a closure system F, where ∑J the set of implications P → P∑F with |P| = 1, and ∑↓F the set of implications P → P∑F with |P| 1. Implications in ∑JF can be computed efficiently from the set of meet-irreducible M(F): but the problem is open for ∑↓F. Many existing algorithms build F as an intermediate step. In this paper, we characterize the cover relation in the family C↓(F) with the same ∑↓, when ordered under set-inclusion. We also show that M(F⊥) the set of meet-irreducible elements of a minimal closure system in C↓(F) can be computed from M(F) in polynomial time for any F in C↓(F). Moreover, the size of M(F⊥) is less or equal to the size of M(F). © Springer-Verlag Berlin Heidelberg 2006.