EMBEDDING FINITELY GENERATED ABELIAN LATTICE-ORDERED GROUPS: HIGMAN'S THEOREM AND A REALISATION OF $\pi$

Abstract

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 $\ell$ -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. T HEOREM . Every finitely generated Abelian lattice-ordered group that has finite rank and a recursively enumerable set of defining relations can be $\ell$ - embedded in a finitely presented lattice-ordered group . If $\xi$ is a real number, let $D(\xi)$ be the Abelian rank 2 group $\Z^2$ with order $(m,n)>0$ if and only if $m+n\xi>0$ . C OROLLARY . $D(\xi)$ can be $\ell$ - embedded in a finitely presented lattice-ordered group if and only if $\xi$ is a recursive real number . Thus an algebraic characterisation of recursive real numbers is obtained. In particular, $\pi$ is ‘ $\ell$ -algebraic’ in that it can be captured by finitely many relations in this language.