Consider a Guigues-Duquenne base of a closure system , where the set of implications with |P|=1, and the set of implications with |P| >1. Implications in can be computed efficiently from the set of meet-irreducible $\mathcal{M}(\mathcal{F})$ ; but the problem is open for . Many existing algorithms build 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 in $\mathcal{C}_{\downarrow}(\mathcal{F})$ . Moreover, the size of $\mathcal{M}(\mathcal{F}_{\perp})$ is less or equal to the size of $\mathcal{M}(\mathcal{F})$ .