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Some examples of hyperarchimedean lattice-ordered groups

Anthony W. Hager;  Chawne M. Kimber — 2004

Fundamenta Mathematicae

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:...

Unique a -closure for some ℓ -groups of rational valued functions

Anthony W. Hager;  Chawne M. Kimber;  Warren W. McGovern — 2005

Czechoslovak Mathematical Journal

Usually, an abelian ℓ -group, even an archimedean ℓ -group, has a relatively large infinity of distinct a -closures. Here, we find a reasonably large class with unique and perfectly describable a -closure, the class of archimedean ℓ -groups with weak unit which are “ ℚ -convex”. ( ℚ is the group of rationals.) Any C ( X , ℚ ) is ℚ -convex and its unique a -closure is the Alexandroff algebra of functions on X defined from the clopen sets; this is sometimes C ( X ) .

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