Epimorphic adjunction of a weak order unit to an Archimedean lattice-ordered group

Abstract

It is shown that an archimedean l l -group G G can be embedded into another, H H , which has a weak unit, by an embedding that is epimorphic in archimedean l l -groups if and only if there is countable A ⊆ G A \subseteq G with A ⊥ = ( 0 ) {A^ \bot } = (0) . Then the extension H H can always be chosen conditionally and laterally σ \sigma -complete and the embedding essential, but can never be generated by G G together with finitely many extra elements unless G G already had a weak unit.