Matroids satisfy a generalized basis exchange where for X and Y in the symmetric difference of bases A and B there exist U and V containing them with |U|=|V| at most rank(X+Y) such that A-U+V and B+U-V are bases, plus a framework for Grassmann-Plücker extensions in characteristic-zero representable
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Existence of ex ante stable and fair fractional allocations in discrete-concave two-sided markets, exactly characterized as Alkan-Gale stable outcomes via concave closures and symmetric convex tie-breaking, with decomposition into lotteries over stable deterministic allocations; extended to ordinal
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Generalizing the Multiple Exchange Property for Matroid Bases
Matroids satisfy a generalized basis exchange where for X and Y in the symmetric difference of bases A and B there exist U and V containing them with |U|=|V| at most rank(X+Y) such that A-U+V and B+U-V are bases, plus a framework for Grassmann-Plücker extensions in characteristic-zero representable