Lattice-ordered pregroups are semi-distributive

We prove that the lattice reduct of every lattice-ordered pregroup is semidistributive. This is a consequence of a certain weak form of the distributive law which holds in lattice-ordered pregroups.

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Published in:	Algebra universalis Vol. 82; no. 1
Main Authors:	Galatos, Nick, Jipsen, Peter, Kinyon, Michael, Přenosil, Adam
Format:	Journal Article
Language:	English
Published:	Cham Springer International Publishing 01-02-2021 Springer Nature B.V
Subjects:	Algebra Mathematics Mathematics and Statistics 06B99 06F05 Residuated lattices Pregroups
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Abstract	We prove that the lattice reduct of every lattice-ordered pregroup is semidistributive. This is a consequence of a certain weak form of the distributive law which holds in lattice-ordered pregroups.
AbstractList	We prove that the lattice reduct of every lattice-ordered pregroup is semidistributive. This is a consequence of a certain weak form of the distributive law which holds in lattice-ordered pregroups.
ArticleNumber	16
Author	Jipsen, Peter Přenosil, Adam Galatos, Nick Kinyon, Michael
Author_xml	– sequence: 1 givenname: Nick surname: Galatos fullname: Galatos, Nick organization: Department of Mathematics, University of Denver – sequence: 2 givenname: Peter surname: Jipsen fullname: Jipsen, Peter organization: Mathematics Chapman University – sequence: 3 givenname: Michael surname: Kinyon fullname: Kinyon, Michael organization: Department of Mathematics, University of Denver – sequence: 4 givenname: Adam surname: Přenosil fullname: Přenosil, Adam email: adam.prenosil@vanderbilt.edu organization: Department of Mathematics, Vanderbilt University
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References	Buszkowski, W.: Lambek grammars based on pregroups. In: P. de Groote, G. Morrill, C. Retoré (eds.) Logical aspects of computational linguistics, LACL 2001, Lecture Notes in Computer Science, vol. 2099, pp. 95–109. Springer (2001) Lambek, J.: Type grammar revisited. In: A. Lecomte, F. Lamarche, G. Perrier (eds.) Logical aspects of computational linguistics, Lecture Notes in Computer Science, vol. 1582, pp. 1–27. Springer (1999) GalatosNJipsenPPeriodic lattice-ordered pregroups are distributiveAlgebra Univ.2012681–2145150300874210.1007/s00012-012-0199-7 Freese, R., Nation, J.: A simple semidistributive lattice Int. J. Algebra Comput. https://doi.org/10.1142/S0218196721500119 LambekJType grammars as pregroupsGrammars20014213910.1023/A:1011444711686 GalatosNJipsenPKowalskiTOnoHResiduated lattices: an algebraic glimpse and substructural logics, Studies in Logic and the Foundations of Mathematics2007New YorkElsevier1171.03001 Buszkowski, W.: Pregroups: models and grammars. In: H.C.M. de Swart (ed.) Relational Methods in Computer Science, RelMiCS 2001, Lecture Notes in Computer Science, vol. 2561, pp. 35–49. Springer (2002) LambekJSome lattice models of bilinear logicAlgebra Univ.199534541550135748310.1007/BF01181877 BuszkowskiWType logic and pregroupsStud. Logic.2007872–3145169236545710.1007/s11225-007-9083-4 GalatosNTsinakisCGeneralized MV-algebrasJ. Algebra20052831254291210208310.1016/j.jalgebra.2004.07.002 N Galatos (703_CR6) 2007 703_CR2 J Lambek (703_CR10) 2001; 4 W Buszkowski (703_CR3) 2007; 87 703_CR1 703_CR4 N Galatos (703_CR5) 2012; 68 N Galatos (703_CR7) 2005; 283 J Lambek (703_CR8) 1995; 34 703_CR9
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Title	Lattice-ordered pregroups are semi-distributive
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