Some examples of hyperarchimedean lattice-ordered groups

Anthony W. Hager¹ ; Chawne M. Kimber²

¹Department of Mathematics and Computer Science Wesleyan University Middletown, CT 06459, U.S.A.
²Department of Mathematics Lafayette College Easton, PA 18042, U.S.A.

Fundamenta Mathematicae, Tome 182 (2004) no. 2, pp. 107-122

Résumé

All $\ell $-groups shall be abelian. An $a$-extension of an $\ell $-group is an extension preserving the lattice of ideals; an $\ell $-group with no proper $a$-extension is called $a$-closed. A hyperarchimedean $\ell $-group is one for which each quotient is archimedean. This paper examines hyperarchimedean $\ell $-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-$\ell $-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.

Détail

DOI : 10.4064/fm182-2-2

Keywords: ell groups shall abelian a extension ell group extension preserving lattice ideals ell group proper a extension called a closed hyperarchimedean ell group which each quotient archimedean paper examines hyperarchimedean ell groups unit their a extensions means yosida representation focussing several previously problems paul conrad asked a closed ideal a closed hyperarchimedean sub ell group product reals f ring which generates hyperarchimedean marlow anderson conrad asked refining first question above a closed minimal prime a closed a closed hyperarchimedean prime isomorphic reals here introduce techniques a extension construct several parameter family examples adjusting parameters provides answers questions above