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Algebra universalis

, 80:54 | Cite as

Distributive laws in residuated binars

  • Wesley FussnerEmail author
  • Peter Jipsen
Article
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Abstract

In residuated binars there are six non-obvious distributivity identities of \(\cdot ,/,\backslash \) over \(\wedge , \vee \). We show that in residuated binars with distributive lattice reducts there are some dependencies among these identities; specifically, there are six pairs of identities that imply another one of these identities, and we provide counterexamples to show that no other dependencies exist among these.

Keywords

Residuated lattices Residuated binars Residuation Subvariety lattices 

Mathematics Subject Classification

06F05 03G10 08B15 

Notes

References

  1. 1.
    Blount, K.: On the Structure of Residuated Lattices. Ph.D. Dissertation, Vanderbilt University, Nashville (1999)Google Scholar
  2. 2.
    Blount, K., Tsinakis, C.: The structure of residuated lattices. Int. J. Algebra Comput. 13, 437–461 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Conradie, W., Palmigiano, A.: Algorithmic correspondence and canonicity for distributive modal logic. Ann. Pure Appl. Log. 163, 338–376 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Fussner, W., Palmigiano, A.: Residuation algebras with functional duals. In: Adaricheva, K. et al. (eds.) Algebras and Lattices in Hawai’i: Honoring Ralph Freese, Bill Lampe, and J.B. Nation, Lulu, pp. 39–46 (2018)Google Scholar
  5. 5.
    Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated lattices: an algebraic glimpse at substructural logics. In: Studies in Logic and the Foundations of Mathematics, vol. 151, pp. xxi+509. Elsevier (2007)Google Scholar
  6. 6.
    Jipsen, P., Jónsson, B., Rafter, J.: Adjoining units to residuated Boolean algebras. Algebra Universalis 34, 118–127 (1995)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DenverDenverUSA
  2. 2.Mathematics Chapman UniversityOrangeUSA

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