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.
Highlights
The notion of closure system or the equivalent notions of closure operator or of complete implicational system are fundamental since they very often appear in pure and applied mathematics
In particular we have characterized the covering relation ≺ of each one of these lattices. This allows us to determine the changes that occur in the join-irreducible elements and the meet-irreducible elements of the lattices C and C ′ when C ≺ C ′ and when we go from C to C ′
In [7], the authors have studied the set of all closure systems having the same poset of join-irreducible elements
Summary
The notion of closure system ( called Moore family) or the equivalent notions of closure operator or of complete implicational system are fundamental since they very often appear in pure and applied mathematics. The changes in a full system of dependencies can be studied in terms of changes in a closure system (see for instance [8] and [14]) It has been known for a long time that the set M of all closure systems defined on a finite set S and ordered by inclusion is a lattice†. An essential task consists in characterizing the covering relation of these lattices; this problem, easy in M , can be difficult in other cases It is related — but not equivalent — to the following one: if L is a class of lattices and L a lattice of L, what are the minimal sets of elements of L which can be deleted from L (respectively, added to L) in order to obtain a lattice which still belongs to L ? The basic tools used in this paper to get our results are the notions of quasi-closed set and of C admissible quasi-closed set, where C is a given class of closure systems
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