Modular elimination in matroids and oriented matroids
Emanuele Delucchi
TL;DR
This paper presents a new axiomatization of matroid theory focusing on modular pairs of circuits, which simplifies and strengthens the axioms, and extends these ideas to oriented matroids.
Contribution
It introduces a novel axiomatization based on modular pairs, providing a cryptomorphic reformulation and enhancing the circuit axioms for oriented matroids.
Findings
A new axiomatization requiring elimination only among modular pairs of circuits
A cryptomorphic reformulation using Crapo's axioms for flats
Strengthened circuit axioms for oriented matroids
Abstract
We introduce a new axiomatization of matroid theory that requires the elimination property only among modular pairs of circuits, and we present a cryptomorphic phrasing thereof in terms of Crapo's axioms for flats. This new point of view leads to a corresponding strengthening of the circuit axioms for oriented matroids.
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