The notion of convex set for subsets of lattices in one particular case was introduced in [1], where it was used to study Paretto's principle in the theory of group choice. This notion is based on a betweenness relation due to Glivenko [2]. Betweenness is used very widely in lattice theory as basis for lattice geometry (see [3], and, especially [4 part 1]).
In the present paper the relative notions of convexity are considered for subsets of an arbitrary lattice.
In section 1 certain relative notions are introduced and studied. The main result is the statement that distributivity is the necessary and sufficient condition for the existence of a variety of natural geometric notions in subsets of a lattice which lead to the definition of convexity.
The study of a variety of notions relating to convexity in subsets is the aim of section 2. In the geometry of convex sets one of the most important results is the description of a convex set by means of its extreme points. One can consider theorem 5 -the main result of this paper- as analog of this geometrical fact.
Two examples are considered in the concluding section.
@article{urn:eudml:doc:38828,
title = {Convexity in subsets of lattices.},
journal = {Stochastica},
volume = {4},
year = {1980},
pages = {129-140},
zbl = {0465.06006},
mrnumber = {MR0599137},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:38828}
}
Ovchinnikov, Sergei V. Convexity in subsets of lattices.. Stochastica, Tome 4 (1980) pp. 129-140. http://gdmltest.u-ga.fr/item/urn:eudml:doc:38828/