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Some classes of partially
ordered rings are studied by means of representations in C(X), where X is a compact
Hausdorff space. The first theorem generalizes the characterizations, due,
respectively, to Harrison and the author, of the subrings of the real field,
and the subrings of C(X). Among the many consequences proved are the
following:
1. If A is a simple ring (no two-sided ideals) partially ordered so that (a) if n is a
positive integer and nαj ≧ 0, then x ≧ 0; and (b) for all x in A there exists an integer
n exceeding x; then A is a commutative field.
2. Let F be a field, let P be a conic prime in F whose intersection Pc in the
center C of F is a primitive (AC) cone in C. Then Pc is an Archimedean order in
C.
3. The compact Hausdorff space X admits a base of power at
most 2ℵ0 of open-and-closed sets if and only if C(X) contains a dense
subfield.
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