There is a long list of open questions rooted in the same underlying problem: understanding the structure of bases or common bases of matroids. These conjectures suggest that matroids may possess much stronger structural properties than are currently known. One example is related to cyclic orderings of matroids. A rank-$r$ matroid is called cyclically orderable if its ground set admits a cyclic ordering such that any interval of $r$ consecutive elements forms a basis. In this paper, we show that if the ground set of a split matroid decomposes into pairwise disjoint bases, then it is cyclically orderable. This result answers a conjecture of Kajitani, Ueno, and Miyano in a special case, and also strengthens Gabow's conjecture for this class of matroids. Our proof is algorithmic, hence it provides a procedure for determining a cyclic ordering in question using a polynomial number of independence oracle calls.

翻译：存在一系列根植于同一基本问题的开放性问题：理解拟阵基或公共基的结构。这些猜想表明，拟阵可能具有比目前已知更强得多的结构性质。其中一个例子与拟阵的循环序有关。一个秩为$r$的拟阵被称为可循环排序的，如果其基础集允许一个循环排序，使得任意$r$个连续元素构成的区间形成一个基。在本文中，我们证明，如果一个分裂拟阵的基础集可以分解为两两不相交的基，那么它是可循环排序的。这一结果在一个特殊情况下回答了Kajitani、Ueno和Miyano的猜想，并且也加强了针对此类拟阵的Gabow猜想。我们的证明是算法性的，因此它提供了一种使用多项式次独立性预言调用来确定所需循环排序的过程。