Rings of almost everywhere defined functions

Abstract

The following representation theorem is proven: A partially ordered commutative ring $R$ is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space $X$ if and only if $R$ is archimedean and localizable. Here we assume that the positive cone of $R$ is closed under multiplication and stable under multiplication with squares, but actually one of these assumptions implies the other. An almost everywhere defined function on $X$ is one that is defined on a dense open subset of $X$. A partially ordered commutative ring $R$ is archimedean if the underlying additive partially ordered abelian group is archimedean, and $R$ is localizable essentially if its order is compatible with the construction of a localization with sufficiently large, positive denominators. As applications we discuss the $\sigma$-bounded case, lattice-ordered commutative rings ($f$-rings), partially ordered fields, and commutative operator algebras.