On the Quotient-Lift Matroid Relation

TL;DR

This paper simplifies the proof of a fundamental theorem relating lift matroids and circuits, and introduces a discrete homotopy theorem connecting matroids of different ranks on the same ground set.

Contribution

It provides a simpler proof of the quotient-lift relation and presents a new discrete homotopy theorem for matroids of varying ranks.

Findings

Simplified proof of the quotient-lift matroid relation

Introduction of a discrete homotopy theorem for matroids of different ranks

Enhanced understanding of matroid circuit unions

Abstract

It is well known that a matroid L is a lift of a matroid M if and only if every circuit of L is the union of some circuits of M. In this paper we give a simpler proof of this important theorem. We also described a discrete homotopy theorem on two matroids of different ranks on the same ground set.