# Convexity in subsets of lattices.

Stochastica (1980)

• Volume: 4, Issue: 2, page 129-140
• ISSN: 0210-7821

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## Abstract

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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.

## How to cite

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Ovchinnikov, Sergei V.. "Convexity in subsets of lattices.." Stochastica 4.2 (1980): 129-140. <http://eudml.org/doc/38828>.

@article{Ovchinnikov1980,
abstract = {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.},
author = {Ovchinnikov, Sergei V.},
journal = {Stochastica},
keywords = {Conjuntos convexos; Subconjuntos; Retículos; convex set; subsets of lattices; betweenness relation},
language = {eng},
number = {2},
pages = {129-140},
title = {Convexity in subsets of lattices.},
url = {http://eudml.org/doc/38828},
volume = {4},
year = {1980},
}

TY - JOUR
AU - Ovchinnikov, Sergei V.
TI - Convexity in subsets of lattices.
JO - Stochastica
PY - 1980
VL - 4
IS - 2
SP - 129
EP - 140
AB - 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.
LA - eng
KW - Conjuntos convexos; Subconjuntos; Retículos; convex set; subsets of lattices; betweenness relation
UR - http://eudml.org/doc/38828
ER -

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