A. C. M. van Rooij

Almost f-algebras: Structure and the Dedekind completion

We prove a representation theorem for almost f-algebras, from which we infer the existence of almost f-algebra multiplications on the Dedekind completions of almost f-algebras.

Whales and the universal completion

In this paper we introduce whales in the collection of subsets of the Boolean algebra of bands in a Dedekind complete Riesz space and employ them to give a short (and constructive) proof of the existence of universal completions for Archimedean Riesz spaces. The existence of a Dedekind completion for an Archimedean Riesz space, emulating the construction of the real numbers from the rationals, in essence goes back to Dedekind. The construction, using Dedekind cuts, is valid in Zermelo-Fraenkel set theory. A Riesz space is called laterally complete if each of its disjoint subsets has a least upper bound. The existence of a lateral completion for Archimedean Riesz spaces can be traced to Maeda and Ogasawara. A laterally complete space which at the same time is Dedekind complete is called universally complete. Fremlin in [9], for good reason naming universally complete spaces inextensible, suggested that the existence of a universal completion could be proved without resource to the MaedaOgasawara Representation Theorem. Zaanen, differently than expected from Fremlin’s paper, followed suit in [12]. Zaanen does not use the Maeda-Ogasawara Representation Theorem, but he does use the Axiom of Choice. In fact, in spite of the elegance of Zaanen’s approach, it falls outside the scope of Zermelo-Fraenkel set theory on many counts and uses the Axiom of Choice repeatedly. We tried to trail Fremlin’s hint more closely and use his dominable subsets, but it was not clear to us how to avoid the Axiom of Choice that way either. In this note we give a different proof which doesn’t use representation theorems and is valid in Zermelo-Fraenkel set theory. Moreover, our proof is easier and uses the novel notion of whales , which might be useful in other circumstances. We add to this that the new aspects of our construction (an easy proof instead of a difficult one and the fact that it is AC-free) gain some significance by the fact that the topic of universal completions has gathered momentum by various recent uses in the theory of positive operators, which on its turn shows promise in applications as diverse as economic theories, dynamical systems and partial differential equations. By the first remark of our paper, we can restrict ourselves without loss of generality to Dedekind complete Riesz spaces. In the following E is a Dedekind complete Riesz space. For unexplained terminology we refer to [1], [2], [3] or [10]. Received by the editors January 10, 1994 and, in revised form, April 4, 1994. 1991 Mathematics Subject Classification. Primary 46A40. Both authors acknowledge partial support from NATO grant 940605. c ©1996 American Mathematical Society 423 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 424 GERARD BUSKES AND ARNOUD VAN ROOIJ 1. Whales Let B be the Boolean algebra of all bands of E. For e ∈ E and B ∈ B we denote the component of e in B by eB. A subset A of B is called a whale if (1) if A ∈ A, B ∈ B and B ⊂ A, then B ∈ A and (2) supA = 1 (= E). The following easy lemma will often be used implicitly. Lemma 1. Every union of whales is a whale and if A and A′ are whales, then so is A∩A′ (= {A ∩A′ : A ∈ A, A′ ∈ A′}). For a whale A we define

Integrals for functions with values in a partially ordered vector space

We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure space leads to the classical space of integrable functions.

The dedekind σ-completion of a Riesz space

Abstract For an Archimedean Riesz space we introduce a Dedekind σ-complete hull, different from Quinns σ-completion and admitting extension of σ-homomorphisms rather than normal homomorphisms.

M-seminorms on spaces of continuous functions

Abstract Let U be a realcompact completely regular Hausdorff space, C ( U ) the vector lattice of all continuous functions on U . We consider representations of M -seminorms on C ( U ) and some subspaces) by semicontinuous functions on U .

Order isomophisms between Riesz spaces

The first aim of this paper is to give a description of the (not necessarily linear) order isomorphisms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}

The Stone-Čech Compactification

C(X)\rightarrow C(Y)

Metrizable Compact Spaces

\end{document}C(X)→C(Y) where X, Y are compact Hausdorff spaces. For a simple case, suppose X is metrizable and T is such an order isomorphism. By a theorem of Kaplansky, T induces a homeomorphism \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}