Independent Spaces of q-Polymatroids

TL;DR

This paper investigates the properties of independent spaces in q-polymatroids, revealing their similarities to q-matroids and establishing new cryptomorphic characterizations despite differences in rank and dimension.

Contribution

It introduces the concept that independent spaces in q-polymatroids share key properties with q-matroids and derives a cryptomorphism based on rank values.

Findings

Independent spaces satisfy properties similar to q-matroids.

Rank values of independent spaces determine the q-polymatroid.

New notions of bases and spanning spaces are developed.

Abstract

This paper is devoted to the study of independent spaces of q-polymatroids. With the aid of an auxiliary q-matroid it is shown that the collection of independent spaces satisfies the same properties as for q-matroids. However, in contrast to q-matroids, the rank value of an independent space does not agree with its dimension. Nonetheless, the rank values of the independent spaces fully determine the q-polymatroid, and this fact can be exploited to derive a cryptomorphism of q-polymatroids. Finally, the notions of minimal spanning spaces, maximally strongly independent spaces, and bases will be elaborated on.