Finitely generated lattice-ordered groups with soluble word problem

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-or...

Finitely Presented Abelian Lattice-Ordered Groups

We give necessary and sufficient conditions for the first-order theory of a finitely presented abelian lattice-ordered group to be decidable. We also show that if the number of generators is at most 3, then elementary equivalence implies isomorphism. We deduce from our methods that the theory of the free MV -algebra on at least 2 generators is undecidable.

Classification of Finitely Generated Lattice-ordered Abelian Groups with Order-unit

A unital l-group (G,u) is an abelian group G equipped with a translation-invariant lattice-order and a distinguished element u, called orderunit, whose positive integer multiples eventually dominate each element of G. We classify finitely generated unital l-groups by sequences W = (W0,W1, . . .) of weighted abstract simplicial complexes, where Wt+1 is obtained from Wt either by the classical Al...

Homological Decision Problems for Finitely Generated Groups with Solvable Word Problem

We show that for finitely generated groups G with solvable word problem, there is no algorithm to determine whether H1(G) is trivial, nor whether H2(G) is trivial.

Finitely Generated Abelian Groups