International Journal of Mathematics and Mathematical Sciences
Volume 24 (2000), Issue 7, Pages 469-479

Ordered groups with greatest common divisors theory

Jiří Močkoř

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.


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 GΓ 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.