  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.

Symbolic Computation