About the Family of Closure Systems Preserving Non-unit Implications in the Guigues-Duquenne Base

Abstract

Consider a Guigues-Duquenne base $\Sigma^{\mathcal{F}} = \Sigma^{\mathcal{F}}_{\mathcal{J}} \cup \Sigma^{\mathcal{F}}_{\downarrow}$ of a closure system $\mathcal{F}$, where $\Sigma_{\mathcal{J}}$ the set of implications $P \rightarrow P^{\Sigma^{\mathcal{F}}}$ with |P|=1, and $\Sigma^{\mathcal{F}}_{\downarrow}$ the set of implications $P \rightarrow P^{\Sigma^{\mathcal{F}}}$ with |P| >1. Implications in $\Sigma^{\mathcal{F}}_{\mathcal{J}}$ can be computed efficiently from the set of meet-irreducible $\mathcal{M}(\mathcal{F})$; but the problem is open for $\Sigma^{\mathcal{F}}_{\downarrow}$. Many existing algorithms build $\mathcal{F}$ as an intermediate step.

In this paper, we characterize the cover relation in the family $\mathcal{C}_{\downarrow}(\mathcal{F})$ with the same Σ_↓, when ordered under set-inclusion. We also show that $\mathcal{M}(\mathcal{F}_{\perp})$ the set of meet-irreducible elements of a minimal closure system in $\mathcal{C}_{\downarrow}(\mathcal{F})$ can be computed from $\mathcal{M}(\mathcal{F})$ in polynomial time for any $\mathcal{F}$ in $\mathcal{C}_{\downarrow}(\mathcal{F})$. Moreover, the size of $\mathcal{M}(\mathcal{F}_{\perp})$ is less or equal to the size of $\mathcal{M}(\mathcal{F})$.