Proof Theory and Ordered Groups
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Ordering theorems, characterizing when partial orders of a group extend to total orders, are used to generate hypersequent calculi for varieties of lattice-ordered groups (\(\ell \)-groups). These calculi are then used to provide new proofs of theorems arising in the theory of ordered groups. More precisely: an analytic calculus for abelian \(\ell \)-groups is generated using an ordering theorem for abelian groups; a calculus is generated for \(\ell \)-groups and new decidability proofs are obtained for the equational theory of this variety and extending finite subsets of free groups to right orders; and a calculus for representable \(\ell \)-groups is generated and a new proof is obtained that free groups are orderable.