Truncated abelian lattice-ordered groups I: The pointed (Yosida) representation

Dans cet article, nous donnons les bases d'une theorie sur la generation des l-groupes et des f-anneaux reduits a partir de certaines sous-structures. Nous sommes concernes en premier lieu par les groupes et anneaux archimediens et hyperarchimediens. Nous discutons le procede d'adjoindre une unite faible a un l-groupe ou une identite a un f-anneau et nous trouvons des differences significatives entre ces situations. Dans les l-groupes, certaines proprietes comme celle d'etre hyperarchimedien ne sont pas toujours transferees aux structures engendrees, comme le montre l'exemple fondamental de Conrad et Martinez, que nous revisitons et elaborons. Par contre, ces proprietes sont transferees dans le cas des f-anneaux reduits.

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Some comments and examples on generation of (hyper-)archimedean $\ell $ -groups and $f$ -rings

In this article, we study essential ideals, socle, and some related ideals of rings of real valued measurable functions. We also study the Goldie dimension of rings of measurable functions.

Essential Ideals in Rings of Measurable Functions

Let A and B be reduced archimedean f-rings, A with identity e; let $A\,\mathop \to \limits^\gamma\,B$ be an l-group homomorphism, and set w = γ (e). We show (with some vagaries of phrasing here) (1) γ = w·ρ for a canonical l-ring homomorphism $A\,\mathop \to \limits^\rho\,B (w)$, where B (w) is an extension of B in which w is a von Neumann regular element, and (2) for XA,XB canonical representation spaces for A, B, γ is realized via composition with a unique partially defined continuous function from XB to XA.

l-group homomorphisms between reduced archimedean f-rings

G always denotes an archimedean `-group. We call G epicom- plete if G is divisible, �-complete, and laterally �-complete. X always denotes a Tychonospace. D (X) is the set of continuous f : X ! (1 ;+1) with f 1 ((1 ;+1)) dense. It is familiar and useful that every G has represen- tations in various D (X): It is known that (A) G is epicomplete with weak unit e if and only if there is a compact basically disconnected space K and an `-group isomorphism GD (K) with e 7! 1K; thus G admits a compatible f-ring multiplication with e as identity. This paper concerns the situation without weak unit and involves this par- J � D (X) means H is an `-group in D (X) situated as (x = 2 closed F =) 9h 2 H with 0 < h(x) < +1 and h(F) = f0g) and (p 2 �X n X and h 2 H =) �h(p) = 0 or �1). A J-representation of G is an `-group isomorphism GH J

A theorem and a question about epicomplete archimedean lattice-ordered groups

In 1962, the author proved that each reduced archimedean f-ring can be represented as an f-ring of continuous extended real-valued function on a locally compact space. The existence of this representation has proved to be quite useful; however, the proof so obscured the definition of the representing functions that deeper applications have remained out of reach. In this paper, we give a new proof of this result; one in which the derivation of the representing functions is more readily accessible. This accessibility is exploited to prove that f-ring homomorphisms on reduced archimedean f-rings are induced by continuous maps between subspaces of their representing spaces, and this leads to further insights into the structure of such homomorphisms.

A representation theorem revisited

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A representation theorem revisited