On the Possibilities of Defining Infinite Oriented Matroids

TL;DR

This paper investigates the challenges of extending finite oriented matroid axioms to infinite cases, demonstrating that key properties like duality and inheritance are not preserved in such extensions.

Contribution

It shows that lifting finite axioms to infinite oriented matroids cannot simultaneously preserve duality and other fundamental properties, highlighting fundamental limitations.

Findings

Circuit axioms do not preserve duality in infinite cases

Orthogonality axioms can preserve duality but lead to proper subclasses

Certain axiom systems cannot be extended without losing key properties

Abstract

Is it possible to define cryptomorphic axiom systems for infinite oriented matroids by lifting some of the axiom systems for finite oriented matroids to the infinite setting while not losing duality in the process? We show that the answer to this question is a twofold "no". First, lifting the circuit axioms neither preserves duality nor inheritance of strong circuit elimination in minors. Second, although duality is kept intact by translating the orthogonality axioms and an axiom system based on the Farkas Lemma, the classes of infinite oriented matroids obtained in this way have the property that one is a proper subclass of the other.