Rings of almost everywhere defined functions

TL;DR

This paper proves a characterization theorem for partially ordered commutative rings as subrings of rings of almost everywhere defined continuous functions on compact spaces, under archimedean and localizability conditions.

Contribution

It establishes a new representation theorem linking algebraic properties of rings with almost everywhere defined functions on compact spaces.

Findings

Characterization of rings as subrings of almost everywhere defined functions

Conditions under which positive cones are closed and stable under multiplication

Applications to lattice-ordered rings, fields, and operator algebras

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.