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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.
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 -