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Four relations on the set of point-hyperplane anti-flags

Mark Pankov, Antonio Pasini

Abstract

There are precisely four arrangements of two point-hyperplane anti-flags. We consider the corresponding relations on the set of such anti-flags and show that each of them can be recovered from any other except in one special case. If the field consists of two elements, then one of the relations cannot be used to recover each of the remaining three. This is related to a bijection between anti-flags and exterior points of the hyperbolic polar space which exists in this case.

Four relations on the set of point-hyperplane anti-flags

Abstract

There are precisely four arrangements of two point-hyperplane anti-flags. We consider the corresponding relations on the set of such anti-flags and show that each of them can be recovered from any other except in one special case. If the field consists of two elements, then one of the relations cannot be used to recover each of the remaining three. This is related to a bijection between anti-flags and exterior points of the hyperbolic polar space which exists in this case.
Paper Structure (9 sections, 20 theorems, 40 equations)

This paper contains 9 sections, 20 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Main objects and results
  3. Proof of the first part of Theorem \ref{['th1']}
  4. Recovering from $\sim_2$ and $\sim_3$
  5. Recovering from $\sim_4$
  6. Recovering from $\sim_1$. The case when $|{\mathbb F}|\ge 4$
  7. Recovering from $\sim_1$. The case when ${\mathbb F}$ is finite
  8. Non-singular points of the hyperbolic form over the field of two elements as anti-flags
  9. Proof of Theorem \ref{['th2']}

Key Result

Theorem 2

The following assertions are fulfilled:

Theorems & Definitions (38)

  • Remark 1
  • Theorem 2
  • Corollary 3
  • proof
  • Remark 4
  • Proposition 5
  • proof
  • Proposition 6
  • Proposition 7
  • Lemma 8
  • ...and 28 more