Filters on a countable vector space

TL;DR

This paper explores combinatorial properties of filters generated by infinite-dimensional subspaces in a countable vector space, drawing analogies to selectivity and stability in ultrafilters.

Contribution

It introduces and analyzes new combinatorial properties of these filters, extending concepts from ultrafilter theory to vector space contexts.

Findings

Identifies key combinatorial properties of filters in vector spaces

Establishes implications between these properties

Draws parallels to ultrafilter properties on natural numbers

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 $\mathrm{FIN}$.