# Some examples of hyperarchimedean lattice-ordered groups

Anthony W. Hager; Chawne M. Kimber

Fundamenta Mathematicae (2004)

- Volume: 182, Issue: 2, page 107-122
- ISSN: 0016-2736

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topAnthony W. Hager, and Chawne M. Kimber. "Some examples of hyperarchimedean lattice-ordered groups." Fundamenta Mathematicae 182.2 (2004): 107-122. <http://eudml.org/doc/283254>.

@article{AnthonyW2004,

abstract = {All ℓ-groups shall be abelian. An a-extension of an ℓ-group is an extension preserving the lattice of ideals; an ℓ-group with no proper a-extension is called a-closed. A hyperarchimedean ℓ-group is one for which each quotient is archimedean. This paper examines hyperarchimedean ℓ-groups with unit and their a-extensions by means of the Yosida representation, focussing on several previously open problems. Paul Conrad asked in 1965: If G is a-closed and M is an ideal, is G/M a-closed? And in 1972: If G is a hyperarchimedean sub-ℓ-group of a product of reals, is the f-ring which G generates also hyperarchimedean? Marlow Anderson and Conrad asked in 1978 (refining the first question above): If G is a-closed and M is a minimal prime, is G/M a-closed? If G is a-closed and hyperarchimedean and M is a prime, is G/M isomorphic to the reals? Here, we introduce some techniques of a-extension and construct a several parameter family of examples. Adjusting the parameters provides answers "No" to the questions above.},

author = {Anthony W. Hager, Chawne M. Kimber},

journal = {Fundamenta Mathematicae},

keywords = {hyperarchimedean lattice-ordered group; Yosida representation theorem; -extension},

language = {eng},

number = {2},

pages = {107-122},

title = {Some examples of hyperarchimedean lattice-ordered groups},

url = {http://eudml.org/doc/283254},

volume = {182},

year = {2004},

}

TY - JOUR

AU - Anthony W. Hager

AU - Chawne M. Kimber

TI - Some examples of hyperarchimedean lattice-ordered groups

JO - Fundamenta Mathematicae

PY - 2004

VL - 182

IS - 2

SP - 107

EP - 122

AB - All ℓ-groups shall be abelian. An a-extension of an ℓ-group is an extension preserving the lattice of ideals; an ℓ-group with no proper a-extension is called a-closed. A hyperarchimedean ℓ-group is one for which each quotient is archimedean. This paper examines hyperarchimedean ℓ-groups with unit and their a-extensions by means of the Yosida representation, focussing on several previously open problems. Paul Conrad asked in 1965: If G is a-closed and M is an ideal, is G/M a-closed? And in 1972: If G is a hyperarchimedean sub-ℓ-group of a product of reals, is the f-ring which G generates also hyperarchimedean? Marlow Anderson and Conrad asked in 1978 (refining the first question above): If G is a-closed and M is a minimal prime, is G/M a-closed? If G is a-closed and hyperarchimedean and M is a prime, is G/M isomorphic to the reals? Here, we introduce some techniques of a-extension and construct a several parameter family of examples. Adjusting the parameters provides answers "No" to the questions above.

LA - eng

KW - hyperarchimedean lattice-ordered group; Yosida representation theorem; -extension

UR - http://eudml.org/doc/283254

ER -

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