Constructions of New q-Cryptomorphisms

TL;DR

This paper establishes multiple equivalent axiomatic systems for q-matroids, extending classical matroid cryptomorphisms to the q-analogue, and highlights key differences between the classical and q-versions.

Contribution

It introduces comprehensive q-matroid axiom systems and demonstrates cryptomorphisms among them, advancing the theoretical framework of q-matroids.

Findings

Established cryptomorphisms for various q-matroid axioms

Highlighted differences between classical and q-matroid theories

Provided a unified set of axioms for q-matroids

Abstract

In the theory of classical matroids, there are several known equivalent axiomatic systems that define a matroid, which are described as matroid cryptomorphisms. A q-matroid is a q-analogue of a matroid where subspaces play the role of the subsets in the classical theory. In this article we establish cryptomorphisms of q-matroids. In doing so we highlight the difference between classical theory and its q-analogue. We introduce a comprehensive set of q-matroid axiom systems and show cryptomorphisms between them and existing axiom systems of a q-matroid. These axioms are described as the rank, closure, basis, independence, dependence, circuit, hyperplane, flat, open space, spanning space, non-spanning space, and bi-colouring axioms.