Advertisement

Semigroup Forum

, Volume 92, Issue 2, pp 361–376 | Cite as

Flat coset decompositions of skew lattices

  • João Pita Costa
  • Karin Cvetko-VahEmail author
Research Article
  • 114 Downloads

Abstract

Skew lattices are non-commutative generalizations of lattices, and the cosets are the building blocks of skew lattices. Every skew lattice embeds into a direct product of a left-handed skew lattice and a right-handed skew lattice. It is therefore natural to consider the flat coset decompositions, i.e. decompositions of a skew lattice into right and left cosets. In the present paper we discuss such decompositions, their properties and the relation to the coset laws for cancellative and symmetric skew lattices.

Keywords

Decomposition Theorem Triangular Matrice Block Form Left Coset Regular Band 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Cornish, W.H.: Boolean skew algebras. Acta Math. Acad. Sci. Hung. 36, 281–291 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cvetko-Vah, K.: Skew lattices of matrices in rings. Algebra Univers. 53, 471–479 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cvetko-Vah, K.: Internal decompositions of skew lattices. Commun. Algebra 35, 243–247 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cvetko-Vah, K.: On the structure of semigroups of idempotent matrices. Linear Algebra Appl. 426, 204–213 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cvetko-Vah, K., Kinyon, M., Leech, J., Spinks, M.: Cancellation in skew lattices. Order 28, 9–32 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cvetko-Vah, K., Pita Costa, J.: On the coset laws for skew lattices. Semigroup Forum 83, 395–411 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fillmore, P., MacDonald, G., Radjabalipour, M., Radjavi, H.: Towards a classification of maximal unicellular bands. Semigroup Forum 49, 195–215 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fillmore, P., MacDonald, G., Radjabalipour, M., Radjavi, H.: Principal ideal bands. Semigroup Forum 59, 362–373 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grätzer, G.: General Lattice Theory. Springer, New York (2003)zbMATHGoogle Scholar
  10. 10.
    Howie, J.M.: Fundamentals of Semigroup Theory. Oxfors science publications, Oxford (1995)zbMATHGoogle Scholar
  11. 11.
    Jordan, P.: Über nichtkommutative Verbände. Arch. Math. 2, 56–59 (1949)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kinyon, M., Leech, J.: Categorical skew lattices. Order 30, 763–777 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kinyon, M., Leech, J., Pita Costa, J.: Distributivity in skew lattices. Semigroup Forum (2015). doi: 10.1007/s00233-015-9722-4
  14. 14.
    Leech, J.: Skew lattices in rings. Algebra Univers. 26, 48–72 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Leech, J.: Normal skew lattices. Semigroup Forum 44, 1–8 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Leech, J.: The geometric structure of skew lattices. Trans. Am. Math. Soc. 335, 823–842 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Petrich, M.: A construction and classification of bands. Math. Nachr. 48, 263–274 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pita Costa, J.: On the coset structure of a skew lattice. Demonstratio Math. 44, 673–692 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Pita Costa, J.: Coset laws for categorical skew lattices. Algebra Univers. 68, 75–89 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pita Costa, J.: On the coset structure of skew lattices, Ph.D. Thesis. University of Ljubljana (2012)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institut Jožef StefanLjubljanaSlovenia
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

Personalised recommendations