Finitely generated lattice-ordered groups with soluble word problem

TL;DR

This paper establishes an analogue of Boone and Higman's theorem for finitely generated lattice-ordered groups, characterizing those with soluble word problem via embeddings into simple and finitely presented lattice-ordered groups.

Contribution

It proves that finitely generated lattice-ordered groups with soluble word problem can be embedded into simple lattice-ordered groups embedded in finitely presented lattice-ordered groups, extending classical results.

Findings

Characterization of soluble word problem in lattice-ordered groups

Embedding theorem for finitely generated lattice-ordered groups

Use of permutation groups in proof

Abstract

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 and the ideas used to prove the lattice-ordered group analogue of Higman's Embedding Theorem.