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Abstract

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.