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

A. W. Hager; D. G. Johnson

doi:10.5802/afst.1276

Some comments and examples on generation of (hyper-)archimedean

ℓ

-groups and

f

-rings

A. W. Hager ¹; D. G. Johnson ²

¹ Department of Mathematics Wesleyan University Middletown, CT, USA, 06459
² 5 W. Oak St. Ramsey, NJ 07446

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. S1, pp. 75-100.

Abstract
Abstract

This paper systematizes some theory concerning the generation of $ℓ$ -groups and reduced $f$ -rings from substructures. We are particularly concerned with archimedean and hyperarchimedean groups and rings. We discuss the process of adjoining a weak order unit to an $ℓ$ -group, or an identity to an $f$ -ring and find significant contrasts between these cases. In $ℓ$ -groups, hyperarchimedeanness and similar properties fail to pass from generating structures to the structures that they generate, as illustrated by a basic example of Conrad and Martinez which we revisit and elaborate. For reduced $f$ -rings, on the other hand, these properties do inherit upwards.

MR

DOI: 10.5802/afst.1276

Author's affiliations:

A. W. Hager ¹; D. G. Johnson ²

¹ Department of Mathematics Wesleyan University Middletown, CT, USA, 06459
² 5 W. Oak St. Ramsey, NJ 07446

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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {75--100},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
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A. W. Hager; D. G. Johnson. Some comments and examples on generation of (hyper-)archimedean $\ell $-groups and $f$-rings. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. S1, pp. 75-100. doi : 10.5802/afst.1276. https://afst.centre-mersenne.org/articles/10.5802/afst.1276/

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