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2004 | 182 | 2 | 107-122
Tytuł artykułu

Some examples of hyperarchimedean lattice-ordered groups

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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.
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  • Department of Mathematics and Computer Science, Wesleyan University, Middletown, CT 06459, U.S.A.
  • Department of Mathematics, Lafayette College, Easton, PA 18042, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-2-2
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