Lexicographic Structures on Vector Spaces

Gutman, A. E. , Emelyanenkov, I. A.
Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 4.

Abstract:
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.

Keywords: maximal cone, dense cone, totally ordered vector space, Archimedean equivalence, Archimedean dominance, lexicographic order, Hamel basis, locally convex space.

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