Cyclic flats of a polymatroid

TL;DR

This paper extends the concept of cyclic flats from matroids to polymatroids, introducing a convolution-like method to construct polymatroids from ranked lattices and measures, facilitating their analysis.

Contribution

It generalizes the characterization of cyclic flats to polymatroids and introduces a novel convolution technique for their construction.

Findings

The ranked lattice of cyclic flats characterizes polymatroids.

A convolution-like method effectively constructs polymatroids from lattices.

Examples demonstrate the practical utility of the convolution technique.

Abstract

Polymatroids can be considered as "fractional matroid" where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a polymatroid carefully, the characterization by Bonin and de Mier of the ranked lattice of cyclic flats carries over to polymatroids. The main tool, which might be of independent interest, is a convolution-like method which creates a polymatroid from a ranked lattice and a discrete measure. Examples show the ease of using the convolution technique.