Vectors of matroids over tracts

TL;DR

This paper extends the theory of matroids over tracts by introducing vectors and covectors, connecting to classical oriented matroids and linear subspaces over fields, thus broadening the mathematical framework.

Contribution

It introduces the notions of vectors and covectors for matroids over tracts, unifying and generalizing existing concepts in matroid theory.

Findings

F-vectors and F-covectors coincide with classical signed vectors in oriented matroids.

F-covectors form a linear subspace of F^E, and F-vectors form its orthogonal complement.

The framework generalizes matroids over fields and oriented matroids within a unified theory.

Abstract

We enrich Baker and Bowler's theory of matroids over tracts with notions of vectors and covectors. In the case of oriented matroids, these $F$-vectors and $F$-covectors coincide with the usual signed vectors and signed covectors. In the case of matroids over a field $F$, the $F$-covector set resp. $F$-vector set of an $F$-matroid is a linear subspace of $F^E$ resp. its orthogonal complement.