Reverse mathematics on lattice ordered groups

Abstract

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 g ≠ e}. 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.