International Journal of Mathematics and Mathematical Sciences
Volume 24 (2000), Issue 7, Pages 469-479
doi:10.1155/S0161171200004087
Ordered groups with greatest common divisors theory
Department of Mathematics, University of Ostrava, Ostrava CZ-702 00, Czech Republic
Received 15 October 1999
Copyright © 2000 Jiří Močkoř. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
An embedding (called a GCD theory) of partly ordered
abelian group G into abelian l-group Γ is investigated
such that any element of Γ is an infimum of a subset
(possible non-finite) from G. It is proved that a GCD theory need
not be unique. A complete GCD theory is introduced and it is
proved that G admits a complete GCD theory if and only if it
admits a GCD theory G→Γ such that Γ is an
Archimedean l-group. Finally, it is proved that a complete GCD
theory is unique (up to o-isomorphisms) and that a po-group
admits the complete GCD theory if and only if any v-ideal is
v-invertible.