Pastures are a class of field-like algebraic objects which include both partial fields and hyperfields and have nice categorical properties. We prove several lift theorems for representations of matroids over pastures, including a generalization of Pendavingh and van Zwam's Lift Theorem for partial fields. By embedding the earlier theory into a more general framework, we are able to establish new results even in the case of lifts of partial fields, for example the conjecture of Pendavingh–van Zwam that their lift construction is idempotent. We give numerous applications to matroid representations, e.g. we show that, up to projective equivalence, every pair consisting of a hexagonal representation and an orientation lifts uniquely to a near-regular representation. The proofs are different from the arguments used by Pendavingh and van Zwam, relying instead on a result of Gelfand–Rybnikov–Stone inspired by Tutte's homotopy theorem.

中文翻译：

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牧场上 matroids 表示的提升定理


牧场是一类类场代数对象，它包括部分场和超场，并且具有良好的分类属性。我们证明了几个用于表示牧场上 matroids 的升力定理，包括 Pendavingh 和 van Zwam 对部分田地的升力定理的推广。通过将早期理论嵌入到更通用的框架中，即使在部分场的升力的情况下，我们也能够建立新的结果，例如 Pendavingh-van Zwam 的猜想，即它们的升力结构是幂等的。我们给出了 matroid 表示的许多应用，例如，我们表明，在射影等价性之前，由六边形表示和方向组成的每对都唯一地提升到近乎规则的表示。这些证明与 Pendavingh 和 van Zwam 使用的论点不同，而是依赖于受 Tutte 同伦定理启发的 Gelfand-Rybnikov-Stone 的结果。