## Nathalie Caspard ; Bernard Monjardet - Some lattices of closure systems on a finite set

dmtcs:309 - Discrete Mathematics & Theoretical Computer Science, January 1, 2004, Vol. 6 no. 2 - https://doi.org/10.46298/dmtcs.309
Some lattices of closure systems on a finite set

Authors: Nathalie Caspard ; Bernard Monjardet

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.

Volume: Vol. 6 no. 2
Published on: January 1, 2004
Imported on: March 26, 2015
Keywords: quasi-closed set,Anti-exchange closure operator,closure system,convex geometry,(locally distributive) lattice,quasi-closed set.,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]