Full-text PDF (619 kB) Citations (2)

DOI: https://doi.org/10.22405/2226-8383-2018-22-1-213-224

Abstract: In this paper «The projective geometry over partially ordered skew fields, II» the investigation of properties for partially ordered linear spaces over partially ordered skew fields is prolonged. This investigation was started in part I «The projective geometry over partially ordered skew fields». Derivative lattices associated partially ordered linear spaces over partially ordered skew fields are examined. More exactly, properties of the convex projective geometry $\mathcal{L}$ of a partially ordered linear space ${}_FV$ over a partially ordered skew field $F$ are considered. The convexity of linear subspaces has meaning the Abelian convexity ($ab$-convexity), which is based on the definition of a convex subgroup for a partially ordered group. Second and third theorems of linear spaces order isomorphisms for interpolation linear spaces over partially ordered skew fields are proved. Some theorems are proved for principal linear subspaces of interpolation linear spaces over directed skew fields. The principal linear subspace $I_a$ of a partially ordered linear space ${}_FV$ over a partially ordered skew field $F$ is the smallest $ab$-convex directed linear subspace of linear space ${}_FV$ which contains the positive element $a\in V$. The analog for the third theorem of linear spaces order isomorphisms for principal linear subspaces is demonstrated in interpolation linear spaces over directed skew fields.

Keywords: a partially ordered ring, a partially ordered skew field, a partially ordered linear space, a directed group, a convex subgroup.

Funding agency	Grant number
Moscow Center of Fundamental and Applied Mathematics

Received: 21.12.2020
Accepted: 21.02.2021

Citation: A. V. Mikhalev, E. E. Shirshova, “The projective geometry over partially ordered skew fields, II”, Chebyshevskii Sb., 22:1 (2021), 213–224

Citation in format AMSBIB

Linking options:

https://www.mathnet.ru/eng/cheb998

https://www.mathnet.ru/eng/cheb/v22/i1/p213

Cycle of papers

• The projective geometry over partially ordered skew fields
  A. V. Mikhalev, E. E. Shirshova
  Fundam. Prikl. Mat., 2020, 23:2, 231–245
• The projective geometry over partially ordered skew fields, II
  A. V. Mikhalev, E. E. Shirshova
  Chebyshevskii Sb., 2021, 22:1, 213–224

This publication is cited in the following 2 articles:

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles