Some lattices of closure systems on a finite set

Abstract : In this paper we study two lattices of significant particular closure systems on a finite set, namely the union stable closure systems and the convex geometries. Using the notion of (admissible) quasi-closed set and of (deletable) closed set, we determine the covering relation \prec of these lattices and the changes induced, for instance, on the irreducible elements when one goes from C to C' where C and C' are two such closure systems satisfying C \prec C'. We also do a systematic study of these lattices of closure systems, characterizing for instance their join-irreducible and their meet-irreducible elements.
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Nathalie Caspard, Bernard Monjardet. Some lattices of closure systems on a finite set. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2004, 6 (2), pp.163-190. ⟨hal-00959003⟩

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