Abstract
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.
Supported by Swiss National Science Foundation grant 200021_146748 and the EU Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 689176.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Anderson, M.E., Feil, T.H.: Lattice-Ordered Groups: An Introduction. Springer, Heidelberg (1988)
Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, Heidelberg (1981)
Ciabattoni, A., Galatos, N., Terui, K.: Algebraic proof theory for substructural logics: cut-elimination and completions. Ann. Pure Appl. Logic 163(3), 266–290 (2012)
Ciabattoni, A., Galatos, N., Terui, K.: Algebraic proof theory: hypersequents and hypercompletions. Ann. Pure Appl. Logic 168(3), 693–737 (2017)
Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic foundations of many-valued reasoning. Kluwer, Berlin (1999)
Clay, A., Smith, L.H.: Corrigendum to [19]. J. Symb. Comput. 44(10), 1529–1532 (2009)
Dantzig, G.B.: Linear Programming and Extensions. Princeton University, Press (1963)
Fuchs, L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford (1963)
Galatos, N., Metcalfe, G.: Proof theory for lattice-ordered groups. Ann. Pure Appl. Logic 8(167), 707–724 (2016)
Holland, W.C.: The lattice-ordered group of automorphisms of an ordered set. Mich. Math. J. 10, 399–408 (1963)
Holland, W.C.: The largest proper variety of lattice-ordered groups. Proc. Am. Math. Soc. 57, 25–28 (1976)
Holland, W.C., McCleary, S.H.: Solvability of the word problem in free lattice-ordered groups. Houston J. Math. 5(1), 99–105 (1979)
Jipsen, P., Montagna, F.: Embedding theorems for classes of GBL-algebras. J. Pure Appl. Algebra 214(9), 1559–1575 (2010)
Kopytov, V.M., Medvedev, N.Y.: The Theory of Lattice-Ordered Groups. Kluwer, Alphen aan den Rijn (1994)
Metcalfe, G., Olivetti, N., Gabbay, D.: Sequent and hypersequent calculi for abelian and Łukasiewicz logics. ACM Trans. Comput. Log. 6(3), 578–613 (2005)
Metcalfe, G., Olivetti, N., Gabbay, D.: Proof Theory for Fuzzy Logics. Springer, Heidelberg (2008)
Montagna, F., Tsinakis, C.: Ordered groups with a conucleus. J. Pure Appl. Algebra 214(1), 71–88 (2010)
Neumann, B.H.: On ordered groups. Am. J. Math. 71(1), 1–18 (1949)
Smith, L.H.: On ordering free groups. J. Symb. Comput. 40(6), 1285–1290 (2005)
Terui, K.: Which structural rules admit cut elimination? – An algebraic criterion. J. Symbolic Logic 72(3), 738–754 (2007)
Weinberg, E.C.: Free abelian lattice-ordered groups. Math. Ann. 151, 187–199 (1963)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer-Verlag GmbH Germany
About this paper
Cite this paper
Colacito, A., Metcalfe, G. (2017). Proof Theory and Ordered Groups. In: Kennedy, J., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2017. Lecture Notes in Computer Science(), vol 10388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55386-2_6
Download citation
DOI: https://doi.org/10.1007/978-3-662-55386-2_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-55385-5
Online ISBN: 978-3-662-55386-2
eBook Packages: Computer ScienceComputer Science (R0)