#### Title

Reverse mathematics on lattice ordered groups

#### Date of Completion

January 2007

#### Keywords

Mathematics

#### Degree

Ph.D.

#### 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 ACA_{0}. 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. ^

#### Recommended Citation

Rogalski, Alexander S, "Reverse mathematics on lattice ordered groups" (2007). *Doctoral Dissertations*. AAI3265795.

http://opencommons.uconn.edu/dissertations/AAI3265795