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Semigroup Forum

, Volume 91, Issue 2, pp 378–400 | Cite as

Distributivity in skew lattices

  • Michael Kinyon
  • Jonathan LeechEmail author
  • João Pita Costa
Research Article

Abstract

Distributive skew lattices satisfying \(x\wedge (y\vee z)\wedge x = (x\wedge y\wedge x) \vee (x\wedge z\wedge x)\) and its dual are studied, along with the larger class of linearly distributive skew lattices, whose totally preordered subalgebras are distributive. Linear distributivity is characterized by the behavior of the natural partial order \(\ge \) on elements in chains of comparable \({\mathcal {D}}\)-classes, \(A>B>C\), with particular attention given to midpoints \(b\) of chains \(a > b > c\) where \(a \in A\), \(b \in B\) and \(c \in C\). Since distributive skew lattices are linearly distributive and have distributive maximal lattice images (but not conversely in general), we give criteria that guarantee that skew lattices with both properties are distributive. In particular symmetric skew lattices (where \(x\wedge y = y\wedge x\) if and only if \(x\vee y = y\vee x\)) that have both properties are distributive.

Keywords

Skew lattice Distributive Partial ordering \({\mathcal {D}}\)-class 

Notes

Acknowledgments

The authors express their appreciation to the referee for a careful reading of the manuscript. The referee’s helpful suggestions have resulted in an improved presentation. The author JPC would like to acknowledge that his work was funded by the EU project TOPOSYS (FP7-ICT-318493-STREP).

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Michael Kinyon
    • 1
  • Jonathan Leech
    • 2
    Email author
  • João Pita Costa
    • 3
  1. 1.University of DenverDenverUSA
  2. 2.Westmont CollegeSanta BarbaraUSA
  3. 3.Inštitut Jozef ŠtefanLjubljanaSlovenia

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