Alternatives for the $q$-matroid axioms of independent spaces, bases, and spanning spaces

TL;DR

This paper refines the axiomatic framework for q-matroids, reducing the number of axioms needed to describe independent spaces, bases, and spanning spaces, and establishes cryptomorphisms between key structures.

Contribution

It introduces two alternative three-axiom descriptions for q-matroid spaces and demonstrates cryptomorphisms linking independent spaces with circuits and bases.

Findings

Reduced the axioms for q-matroids to three using two alternative approaches.

Established cryptomorphisms between independent spaces and circuits.

Established cryptomorphisms between independent spaces and bases.

Abstract

It is well known that in q-matroids, axioms for independent spaces, bases, and spanning spaces differ from the classical case of matroids, since the straightforward q-analogue of the classical axioms does not give a q-matroid. For this reason, a fourth axiom has been proposed. In this paper we show how we can describe these spaces with only three axioms, providing two alternative ways to do that. As an application, we show direct cryptomorphisms between independent spaces and circuits and between independent spaces and bases.