# Extension-lifting Bijections for Oriented Matroids

## TL;DR

This paper introduces a new family of bijections between bases and orientations of oriented matroids, extending geometric bijections for regular matroids, with implications for matroid programming and triangulations.

## Contribution

It generalizes geometric bijections to oriented matroids via circuit and cocircuit signatures from liftings and extensions, providing new characterizations.

## Key findings

- Characterized generic single-element liftings and extensions.
- Established bijections between bases and orientations.
- Discussed implications for oriented matroid programming.

## Abstract

Extending the notion of geometric bijections for regular matroids, introduced by the first and third author with Matthew Baker, we describe a family of bijections between bases of an oriented matroid and special orientations. These bijections are specified by a pair of circuit and cocircuit signatures coming respectively from a generic single-element lifting and extension. We then characterize generic single-element liftings and extensions using these bijections. We also explain the relation of our work with the works of Gioan--Las Vergnas and Ding. Some implications in oriented matroid programming and oriented matroid triangulations are also discussed.