EMBEDDING FINITELY GENERATED ABELIAN LATTICE-ORDERED GROUPS: HIGMAN'S THEOREM AND A REALISATION OF $\pi$
A. M. W. Glass, Vincenzo Marra · Journal of the London Mathematical Society · 2003
Graham Higman proved that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. The analogue for lattice-ordered groups is considered here. Clearly, the finitely generated lattice-ordered groups that can be ℓ-embedded in finitely presented lattice-ordered groups must have recursively enumerable sets of defining relations. The converse direction is proved for a special class of lattice-ordered groups. Theorem. Every finitely generated Abelian lattice-ordered group that has finite rank and a recursively enumerable set of defining relations can be ℓ-embedded in a finitely presented lattice-ordered group. If ξ is a real number, let D(ξ) be the Abelian rank 2 group Z2 with order (m, n)>0 if and only if m+nξ>0. Corollary. D(ξ) can be ℓ-embedded in a finitely presented lattice-ordered group if and only if ξ is a recursive real number. Thus an algebraic characterisation of recursive real numbers is obtained. In particular, π is ‘ℓ-algebraic’ in that it can be captured by finitely many relations in this language.