单元素扩展的偏斜轨上的单元素扩展
Single-element extensions of matroids over skew tracts
AI总结：
本文研究了偏斜轨上单元素扩展的特征化，提出Pathetic Cancellation条件，推广了Crapo和Las Vergnas的结果，并探讨了严苛偏斜超域上的单元素扩展特征。
摘要：
偏斜轨上的单元素扩展提供了一个代数框架，同时泛化了线性子空间、单子、定向单子、相位单子和其他一些『带有额外结构的单子』的概念。一个单元素扩展的单子M over一个偏斜轨T是通过在M上添加一个额外元素得到的单子M~ over T。Crapo以普通单子的单元素扩展为特征，而Las Vergnas以定向单子的单元素扩展为特征，以单元素扩展其秩2收缩来描述这些结果。Crapo和Las Vergnas的结果不能推广到偏斜轨上的单子，但我们将展示一个关于偏斜轨的必要且充分条件，称为Pathetic Cancellation，使得结果可以推广到弱偏斜轨上的单子。严苛的偏斜超域是偏斜轨的一个特殊情况，其在许多方面像偏斜域一样行为。我们找到了强单子在严苛的偏斜超域上的单元素扩展的特征化。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
