Finitely generated lattice-ordered groups with soluble word problem

William W. Boone and Graham Higman proved that a finitely generated group has soluble word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. We prove the exact analogue for lattice-ordered groups: Theorem: A finitely generated lattice-ordered group has soluble word problem if and only if it can be `-embedded in an `-simple lattice-ordered group that can be `-embedded in a finitely presented lattice-ordered group. The proof uses permutation groups, a technique of Holland and McCleary, and the ideas used to prove the lattice-ordered group analogue of Higman’s Embedding Theorem. [Accepted and will appear in J. Group Theory in 2008.] —————————————– AMS Classification: 06F15, 20F60, 20B27.