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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 strongarXiv:1709.09707v2 fatcat:r2ufkj563zhyphwzaltbsgrzka