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Filters on a countable vector space

Iian B. Smythe

Abstract

We study various combinatorial properties, and the implications between them, for filters generated by infinite-dimensional subspaces of a countable vector space. These properties are analogous to selectivity for ultrafilters on the natural numbers and stability for ordered-union ultrafilters on $\mathrm{FIN}$.

Filters on a countable vector space

Abstract

We study various combinatorial properties, and the implications between them, for filters generated by infinite-dimensional subspaces of a countable vector space. These properties are analogous to selectivity for ultrafilters on the natural numbers and stability for ordered-union ultrafilters on .

Paper Structure

This paper contains 6 sections, 21 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $\mathrm{FIN}$ and the finite field of order two
  3. Fullness and maximality
  4. The $(p)$-property and its relatives
  5. The restricted Gowers game and strategic filters
  6. Summary and further questions

Key Result

Theorem 1.4

Let $\mathcal{F}$ be a $(p^+)$-filter on $E$.As mentioned in MR3864398, an apparent weakening of the $(p)$-property akin to semiselectivity, namely that a sequence of dense open subsets of $\mathcal{F}$ must possess a diagonalization in $\mathcal{F}$, is all that is used in the proof of this result.

Figures (2)

  • Figure 1: The implications for block filters proved in MR3864398.
  • Figure 2: Updated implications from Figure \ref{['fig:old_imps']}.

Theorems & Definitions (42)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Theorem 1.1 in MR3864398
  • Definition 1.5
  • Theorem 2.1
  • Theorem 2.2: Theorem 6.3 in MR3864398
  • Theorem 2.3: Theorem 4.2 in MR891244
  • Corollary 2.4
  • proof
  • ...and 32 more