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.