  We present an algebraic framework which simultaneously generalizes the notion
of linear subspaces, matroids, valuated matroids, oriented matroids, and
regular matroids. To do this, we first introduce algebraic objects called
tracts which generalize both hyperfields in the sense of Krasner and partial
fields in the sense of Semple and Whittle. We then define matroids over tracts;
in fact, there are (at least) two natural notions of matroid in this general
context, which we call weak and strong matroids. We give "cryptomorphic" axiom
systems for such matroids in terms of circuits, Grassmann-Pl\"ucker functions,
and dual pairs, and establish some basic duality results. We then explore
sufficient criteria for the notions of weak and strong matroids to coincide.
For example, if $F$ is a particularly nice kind of tract called a doubly
distributive partial hyperfield, we show that the notions of weak and strong
$F$-matroids coincide. We also give examples of tracts $F$ and weak
$F$-matroids which are not strong. Our theory of matroids over tracts is
closely related to, but more general than, "matroids over fuzzy rings" in the
sense of Dress and Dress-Wenzel.
