论文标题
Strassen类型的vergleichsstellensatz，用于通过其分数的半分类的非交换性预定半牙
A Vergleichsstellensatz of Strassen's Type for a Noncommutative Preordered Semialgebra through the Semialgebra of its Fractions
论文作者
论文摘要
预定的半牙布和半肌是经常出现在实际代数几何形状中的两种代数结构，通常在其中起重要作用。他们在实际代数几何，概率理论，理论计算机科学，量子信息理论，\ emph {etc。}领域中具有许多有趣而有希望的应用。在这些应用中，Strassen的Vergleichsstellensatz及其广义版本，这些版本是实际代数几何形状中这些potitivstellensätze的类似物，扮演着重要角色。尽管这些Vergleichsstellensätze仅接受交换性设置（对于所讨论的半段），但在本文中，我们证明了Fritz提出的广义Vergleichsstellensätze之一[\ emph {comm。代数}，49（2）（2021），pp。482-499]。我们证明中最关键的一步是定义非交易性半分类的分数的半分类，该分数概括了文献中的定义。我们的新vergleichsstellensatz在所有单调同型同构中诱导的非交通性半ge骨上的放松预订表征了$ \ mathbb {r} _+$，由其他三个同等条件下的其他等效条件在其派生的预订的半分数上，这可能会导致未来的应用程序。
Preordered semialgebras and semirings are two kinds of algebraic structures occurring in real algebraic geometry frequently and usually play important roles therein. They have many interesting and promising applications in the fields of real algebraic geometry, probability theory, theoretical computer science, quantum information theory, \emph{etc.}. In these applications, Strassen's Vergleichsstellensatz and its generalized versions, which are analogs of those Positivstellensätze in real algebraic geometry, play important roles. While these Vergleichsstellensätze accept only a commutative setting (for the semirings in question), we prove in this paper a noncommutative version of one of the generalized Vergleichsstellensätze proposed by Fritz [\emph{Comm. Algebra}, 49 (2) (2021), pp. 482-499]. The most crucial step in our proof is to define the semialgebra of the fractions of a noncommutative semialgebra, which generalizes the definitions in the literature. Our new Vergleichsstellensatz characterizes the relaxed preorder on a noncommutative semialgebra induced by all monotone homomorphisms to $\mathbb{R}_+$ by three other equivalent conditions on the semialgebra of its fractions equipped with the derived preorder, which may result in more applications in the future.
