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Preordered semialgebras and semirings are two kinds of algebraic structuresoccurring in real algebraic geometry frequently and usually play importantroles therein. They have many interesting and promising applications in thefields of real algebraic geometry, probability theory, theoretical computerscience, quantum information theory, \emph{etc.}. In these applications,Strassen's Vergleichsstellensatz and its generalized versions, which areanalogs of those Positivstellens\"atze in real algebraic geometry, playimportant roles. While these Vergleichsstellens\"atze accept only a commutativesetting (for the semirings in question), we prove in this paper anoncommutative version of one of the generalized Vergleichsstellens\"atzeproposed by Fritz [\emph{Comm. Algebra}, 49 (2) (2021), pp. 482-499]. The mostcrucial step in our proof is to define the semialgebra of the fractions of anoncommutative semialgebra, which generalizes the definitions in theliterature. Our new Vergleichsstellensatz characterizes the relaxed preorder ona noncommutative semialgebra induced by all monotone homomorphisms to$\mathbb{R}_+$ by three other equivalent conditions on the semialgebra of itsfractions equipped with the derived preorder, which may result in moreapplications in the future.

A Vergleichsstellensatz of Strassen's Type for a Noncommutative Preordered Semialgebra through the Semialgebra of its Fractions