Lexicographic structures on vector spaces

A. E. Gutman, I. A. Emelyanenkov

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)42-55
Number of pages14
JournalVladikavkaz Mathematical Journal
Volume21
Issue number4
DOIs
Publication statusPublished - 1 Jan 2019

Keywords

  • Archimedean dominance
  • Archimedean equivalence
  • Dense cone
  • Hamel basis
  • Lexicographic order
  • Locally convex space
  • Maximal cone
  • Totally ordered vector space

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