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

• Alain Gély
• Lhouari Nourine
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3874)

## 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 $$\mathcal{M}(\mathcal{F})$$.

## Keywords

Closure system Guigues-Duquenne base Equivalence relation

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2006

## Authors and Affiliations

• Alain Gély
• 1
• Lhouari Nourine
• 1
1. 1.Université Clermont II Blaise Pascal, LIMOSAubièreFrance