Reverse mathematics on lattice ordered groups

Date of Completion

January 2007






Several theorems about lattice-ordered groups are analyzed. RCA 0 is sufficient to prove the induced order on a quotient of ℓ-groups and the Riesz Decomposition Theorem. WKL 0 is equivalent to the statement "An abelian group G is torsion free if and only if it is lattice-orderable." ACA 0 is equivalent to the existence of various substructures: the join of two convex ℓ-subgroups, the convex closure of an ℓ-subgroup, the polar subgroup X of an ℓ-subgroup X, and a sequence of values {V(g): g ge}. The standard proof of Holland's Embedding Theorem uses ACA0. Holland's Theorem is equivalent to the existence of a sequence of excluding prime subgroups {P(g): g e}, and the existence of such a sequence is provable in WKL 0 when G is abelian. ^