On the Dedekind completion of function rings

This paper introduces the frame of partially defined real numbers and the lattice-ordered ring of partial real functions on a frame. This is then used to construct the order completion of rings of pointfree continuous real functions. The bounded and integer-valued cases are also analysed. The application of this pointfree approach to the classical case C(

DEDEKIND COMPLETIONS OF BOUNDED ARCHIMEDEAN ℓ-ALGEBRAS

All algebras considered in this paper are commutative with 1. Let baℓ be the category of bounded Archimedean ℓ-algebras. We investigate Dedekind completions and Dedekind complete algebras in baℓ. We give several characterizations for A ∈ baℓ to be Dedekind complete. Also, given A, B ∈ baℓ, we give several characterizations for B to be the Dedekind completion of A. We prove that unlike general Gelfand-Neumark-Stone duality, the duality for Dedekind complete algebras does not require any form of the Stone–Weierstrass Theorem. We show that taking the Dedekind completion is not functorial, but that it is functorial if we restrict our attention to those A ∈ baℓ that are Baer rings. As a consequence of our results, we give a new characterization of when A ∈ baℓ is a C*-algebra. We also show that A is a C*-algebra if and only if A is the inverse limit of an inverse family of clean C*-algebras. We conclude the paper by discussing how to derive Gleason's theorem about projective compact Hausdorff spaces and projective covers of compact Hausdorff spaces from our results.

Relatively uniform convergences in archimedean lattice ordered groups

For an archimedean lattice ordered group G let G d and G∧ be the divisible hull or the Dedekind completion of G, respectively. Put G d∧ = X. Then X is a vector lattice. In the present paper we deal with the relations between the relatively uniform convergence on X and the relatively uniform convergence on G. We also consider the relations between the o-convergence and the relatively uniform convergence on G. For any nonempty class τ of lattice ordered groups we introduce the notion of τ-radical class; we apply this notion by investigating relative uniform convergences.