Proof Theory and Ordered Groups

Colacito, Almudena; Metcalfe, George (2017). Proof Theory and Ordered Groups. In: Kennedy, Juliette; de Queiroz, Ruy J.G.B. (eds.) Proceedings of WoLLIC 2017. Logic, Language, Information, and Computation. Lecture Notes in Computer Science: Vol. 10388 (pp. 80-91). Springer-Verlag Berlin Heidelberg 10.1007/978-3-662-55386-2_6

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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 (ℓ-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 ℓ-groups is generated using an ordering theorem for abelian groups; a calculus is generated for ℓ-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 ℓ-groups is generated and a new proof is obtained that free groups are orderable.

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