Geodesic
Lexicographic structures on vector spaces
Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 4, pp. 42-55 
A. E. Gutman; I. A. Emelyanenkov. Lexicographic structures on vector spaces. Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 4, pp. 42-55. http://geodesic.mathdoc.fr/item/VMJ_2019_21_4_a3/
@article{VMJ_2019_21_4_a3,
     author = {A. E. Gutman and I. A. Emelyanenkov},
     title = {Lexicographic structures on vector spaces},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {42--55},
     year = {2019},
     volume = {21},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2019_21_4_a3/}
}
TY  - JOUR
AU  - A. E. Gutman
AU  - I. A. Emelyanenkov
TI  - Lexicographic structures on vector spaces
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2019
SP  - 42
EP  - 55
VL  - 21
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/VMJ_2019_21_4_a3/
LA  - ru
ID  - VMJ_2019_21_4_a3
ER  - 
%0 Journal Article
%A A. E. Gutman
%A I. A. Emelyanenkov
%T Lexicographic structures on vector spaces
%J Vladikavkazskij matematičeskij žurnal
%D 2019
%P 42-55
%V 21
%N 4
%U http://geodesic.mathdoc.fr/item/VMJ_2019_21_4_a3/
%G ru
%F VMJ_2019_21_4_a3

Voir la notice de l'article provenant de la source Math-Net.Ru

Basic properties are described of the Archimedean equivalence and dominance in a totally ordered vector space. The notion of (pre)lexicographic structure on a vector space is introduced and studied. A lexicographic structure is a duality between vectors and points which makes it possible to represent an abstract ordered vector space as an isomorphic space of real-valued functions endowed with a lexicographic order. The notions of function and basic lexicographic structures are introduced. Interrelations are clarified between an ordered vector space and its function lexicographic representation. A new proof is presented for the theorem on isomorphic embedding of a totally ordered vector space into a lexicographically ordered space of real-valued functions with well-ordered supports. A criterion is obtained for denseness of a maximal cone with respect to the strongest locally convex topology. Basic maximal cones are described in terms of sets constituted by pairwise nonequivalent vectors. The class is characterized of vector spaces in which there exist nonbasic maximal cones.

[1] Kutateladze S. S., Fundamentals of Functional Analysis, Kluwer Academic Publishers, Dordrecht etc., 1996, 284 pp. | MR

[2] Fel'dman M. M., “Sublinear Operators Defined on a Cone”, Siberian Mathematical Journal, 16:6 (1975), 1005–1015 | DOI | MR

[3] Vulikh B. Z., Introduction to the Theory of Cones in Normed Spaces, KGU, Kalinin, 1977, 84 pp.

[4] Fuchs L., Partially Ordered Algebraic Systems, Pergamon Press, Oxford, etc., 1963, 229 pp. | MR | Zbl

[5] Hausner M., Wendel J. G., “Ordered vector spaces”, Proc. Amer. Math. Soc., 3 (1952), 977–982 | DOI | MR | Zbl

[6] Wilansky A., Modern Methods in Topological Vector Spaces, McGraw-Hill, New York, 1978 | MR | Zbl

[7] Jech T., Set Theory. The Third Millennium Edition, revised and expanded, Springer, Berlin, etc., 2003 | MR | Zbl