论文标题
偏斜区域上的弱矩形的扩展和严格的偏斜超菲尔德
Extensions of weak matroids over skew tracts and strong matroids over stringent skew hyperfields
论文作者
论文摘要
偏斜区域上的矩形提供了一个代数框架，同时概括了线性子空间，矩阵，定向的矩形，相分子矩阵和其他一些“具有额外结构的矩形”的概念。在偏斜的tract $ t $上的Matroid $ \ Mathcal {M} $的单元素扩展是一种Matroid $ \ widetilde {\ Mathcal {M}} $，超过从$ \ Mathcal {M} $中获得的$ t $，通过添加一个元素。 Crapo以其等级2收缩的单元素扩展来表征普通矩形的单元素扩展，而LAS Vergnas表征了定向矩阵的单元素扩展。 Crapo和Las vergnas的结果并未概括为偏斜段上的矩形，但是我们将在称为可悲的取消的偏斜区域表现出必要且充分的条件，以便该结果可以推广到较弱的矩阵上，而不是偏斜小道。 严格的偏斜Hyperfields是偏斜区域的特殊情况，在许多方面都表现得很偏斜。我们发现，在严格的偏斜超场上，强曲霉的单一元素扩展的表征。
Matroids over skew tracts provide an algebraic framework simultaneously generalizing the notions of linear subspaces, matroids, oriented matroids, phased matroids, and some other "matroids with extra structure". A single-element extension of a matroid $\mathcal{M}$ over a skew tract $T$ is a matroid $\widetilde{\mathcal{M}}$ over $T$ obtained from $\mathcal{M}$ by adding one more element. Crapo characterized single-element extensions of ordinary matroids, and Las Vergnas characterized single-element extensions of oriented matroids, in terms of single-element extensions of their rank 2 contractions. The results of Crapo and Las Vergnas do not generalize to matroids over skew tracts, but we will show a necessary and sufficient condition on skew tracts, called Pathetic Cancellation, such that the result can generalize to weak matroids over skew tracts. Stringent skew hyperfields are a special case of skew tracts which behave in many ways like skew fields. We find a characterization of single-element extensions of strong matroids over stringent skew hyperfields.
