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 Positivstellens\"atze in real algebraic geometry, play important roles. While these Vergleichsstellens\"atze accept only a commutative setting (for the semirings in question), we prove in this paper a noncommutative version of one of the generalized Vergleichsstellens\"atze 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.

翻译：预序半代数和半环是实代数几何中频繁出现的两类代数结构，并通常在其中扮演重要角色。它们在实代数几何、概率论、理论计算机科学、量子信息论等领域具有许多有趣且具有前景的应用。在这些应用中，Strassen比较定理及其推广版本（相当于实代数几何中的正性定理）发挥着重要作用。尽管这些比较定理仅适用于交换性设置（对于所讨论的半环而言），但本文证明了由Fritz [《Comm. Algebra》，49卷(2期)(2021)，第482-499页] 提出的一个推广比较定理的非交换版本。我们证明过程中最关键的一步是定义了非交换半代数的分式半代数，这是对文献中相关定义的推广。我们的新比较定理刻画了由所有到$\mathbb{R}_+$的单调同态诱导的非交换半代数上的松弛预序，并通过其分式半代数（配备导出预序）上的三个等价条件给出了该松弛预序的特征，这一结果未来可能具有更广泛的应用。