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No, not every matroid satisfies this property. For example, it is known to fail for the cycle matroid of $K_4$. The matroids that satisfy your property are called base orderable matroids. There are important classes of matroids that are base orderable, such as transversal matroids. Moreover, base orderability is a minor-closed property, but Ingleton proved that there are actually an infinite number of excluded minors. See these slides of Joseph Bonin for more information. For example, the slides include a proof that $K_4$ is not base orderable.

No, not every matroid satisfies this property. For example, it is known to fail for the cycle matroid of $K_4$. The matroids that satisfy your property are called base orderable matroids. There are important classes of matroids that are base orderable, such as transversal matroids. Moreover, base orderability is a minor-closed property, but Ingleton proved that there are actually an infinite number of excluded minors. See these slides of Joseph Bonin for more information. For example, the slides include a proof that $M(K_4) $ is not base orderable.

No, not every matroid satisfies this property. For example, it is known to fail for the cycle matroid of $K_4$. The matroids that satisfies your property are called base orderable matroids. There are important classes of matroids that are base orderable, such as transversal matroids. Moreover, base orderability is a minor-closed property, and Ingleton proved that there are actually an infinite number of excluded minors. See these slides of Joseph Bonin for more information. For example, the slides include a proof that $K_4$ is not base orderable.

No, not every matroid satisfies this property. For example, it is known to fail for the cycle matroid of $K_4$. The matroids that satisfy your property are called base orderable matroids. There are important classes of matroids that are base orderable, such as transversal matroids. Moreover, base orderability is a minor-closed property, but Ingleton proved that there are actually an infinite number of excluded minors. See these slides of Joseph Bonin for more information. For example, the slides include a proof that $K_4$ is not base orderable.

No, not every matroid satisfies this property. For example, it is known to fail for the cycle matroid of $K_4$. The matroids that satisfies your property are called base orderable matroids. Base orderability is a minor-closed property, and Ingleton proved that there are actually an infinite number of excluded minors. See these slides of Joseph Bonin for more information.

No, not every matroid satisfies this property. For example, it is known to fail for the cycle matroid of $K_4$. The matroids that satisfies your property are called base orderable matroids. There are important classes of matroids that are base orderable, such as transversal matroids. Moreover, base orderability is a minor-closed property, and Ingleton proved that there are actually an infinite number of excluded minors. See these slides of Joseph Bonin for more information. For example, the slides include a proof that $K_4$ is not base orderable.