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Abstract We develop and study a generalization of commutative rings calledbands, along with the corresponding geometric theory ofband schemes. Bands generalize both hyperrings, in the sense of Krasner, and partial fields in the sense of Semple and Whittle. They form a ring‐like counterpart to the field‐like category ofidyllsintroduced by the first and third authors in the previous work. The first part of the paper is dedicated to establishing fundamental properties of bands analogous to basic facts in commutative algebra. In particular, we introduce various kinds of ideals in a band and explore their properties, and we study localization, quotients, limits, and colimits. The second part of the paper studies band schemes. After giving the definition, we present some examples of band schemes, along with basic properties of band schemes and morphisms thereof, and we describe functors into some other scheme theories. In the third part, we discuss some “visualizations” of band schemes, which are different topological spaces that one can functorially associate to a band scheme . 

Thefoundationof a matroid is a canonical algebraic invariant which classifies, in a certain precise sense, all representations of the matroid up to rescaling equivalence. Foundations of matroids arepastures, a simultaneous generalization of partial fields and hyperfields which are special cases of both tracts (as defined by the first author and Bowler) and ordered blue fields (as defined by the second author). Using deep results due to Tutte, Dress–Wenzel, and Gelfand–Rybnikov–Stone, we give a presentation for the foundation of a matroid in terms of generators and relations. The generators are certain “cross-ratios” generalizing the cross-ratio of four points on a projective line, and the relations encode dependencies between cross-ratios in certain low-rank configurations arising in projective geometry. Although the presentation of the foundation is valid for all matroids, it is simplest to apply in the case of matroidswithout large uniform minors. i.e., matroids having no minor corresponding to five points on a line or its dual configuration. For such matroids, we obtain a complete classification of all possible foundations. We then give a number of applications of this classification theorem, for example: We prove the following strengthening of a 1997 theorem of Lee and Scobee: every orientation of a matroid without large uniform minors comes from a dyadic representation, which is unique up to rescaling. For a matroid $M$without large uniform minors, we establish the following strengthening of a 2017 theorem of Ardila–Rincón–Williams: if $M$is positively oriented then $M$is representable over every field with at least 3 elements. Two matroids are said to belong to the samerepresentation classif they are representable over precisely the same pastures. We prove that there are precisely 12 possibilities for the representation class of a matroid without large uniform minors, exactly three of which are not representable over any field. 

Abstract We give a new proof, along with some generalizations, of a folklore theorem (attributed to Laurent Lafforgue) that a rigid matroid (i.e., a matroid with indecomposable basis polytope) has only finitely many projective equivalence classes of representations over any given field.