Advertisement

Algebra universalis

, 80:11 | Cite as

Symmetric embeddings of free lattices into each other

  • Gábor CzédliEmail author
  • Gergő Gyenizse
  • Ádám Kunos
Open Access
Article
  • 53 Downloads

Abstract

By a 1941 result of Ph. M. Whitman, the free lattice \({{\,\mathrm{FL}\,}}(3)\) on three generators includes a sublattice S that is isomorphic to the lattice \({{\,\mathrm{FL}\,}}(\omega )={{\,\mathrm{FL}\,}}(\aleph _0)\) generated freely by denumerably many elements. The first author has recently “symmetrized” this classical result by constructing a sublattice \(S\cong {{\,\mathrm{FL}\,}}(\omega )\) of \({{\,\mathrm{FL}\,}}(3)\) such that S is selfdually positioned in \({{\,\mathrm{FL}\,}}(3)\) in the sense that it is invariant under the natural dual automorphism of \({{\,\mathrm{FL}\,}}(3)\) that keeps each of the three free generators fixed. Now we move to the furthest in terms of symmetry by constructing a selfdually positioned sublattice \(S\cong {{\,\mathrm{FL}\,}}(\omega )\) of \({{\,\mathrm{FL}\,}}(3)\) such that every element of S is fixed by all automorphisms of \({{\,\mathrm{FL}\,}}(3)\). That is, in our terminology, we embed \({{\,\mathrm{FL}\,}}(\omega )\) into \({{\,\mathrm{FL}\,}}(3)\) in a totally symmetric way. Our main result determines all pairs \((\kappa ,\lambda )\) of cardinals greater than 2 such that \({{\,\mathrm{FL}\,}}(\kappa )\) is embeddable into \({{\,\mathrm{FL}\,}}(\lambda )\) in a totally symmetric way. Also, we relax the stipulations on \(S\cong {{\,\mathrm{FL}\,}}(\kappa )\) by requiring only that S is closed with respect to the automorphisms of \({{\,\mathrm{FL}\,}}(\lambda )\), or S is selfdually positioned and closed with respect to the automorphisms; we determine the corresponding pairs \((\kappa ,\lambda )\) even in these two cases. We reaffirm some of our calculations with a computer program developed by the first author. This program is for the word problem of free lattices, it runs under Windows, and it is freely available.

Keywords

Free lattice sublattice Dual automorphism Symmetric embedding Selfdually positioned Totally symmetric embedding Lattice word problem Whitman’s condition FL(3) FL(omega) 

Mathematics Subject Classification

06B25 

Notes

Acknowledgements

Open access funding provided by University of Szeged (SZTE).

References

  1. 1.
    Czédli, G.: On the word problem of lattices with the help of graphs. Period. Math. Hung. 23, 49–58 (1991)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Czédli, G.: A selfdual embedding of the free lattice over countably many generators into the three-generated one. Acta Math. Hung. 148, 100–108 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dean, R.A.: Completely free lattices generated by partially ordered sets. Trans. Am. Math. Soc. 83, 238–249 (1956)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dean, R.A.: Free lattices generated by partially ordered sets and preserving bounds. Can. J. Math. 16, 136–148 (1964)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dilworth, R.P.: Lattices with unique complements. Trans. Am. Math. Soc. 57, 123–154 (1945)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Evans, T.: The word problem for abstract algebras. Lond. Math. Soc. 26, 64–71 (1951)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Freese, R.: Connected Components of the Covering Relation in Free Lattices. Universal Algebra and Lattice Theory (Charleston, S.C., 1984), Lecture Notes in Math., vol. 1149, pp. 82–93. Springer, Berlin (1985)Google Scholar
  8. 8.
    Freese, R.: Free lattice algorithms. Order 3, 331–344 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Freese, R., Ježek, J., Nation, J. B.: Free Lattices. Mathematical Surveys and Monographs, vol. 42. American Mathematical Society, Providence (1995)Google Scholar
  10. 10.
    Freese, R., Nation, J.B.: Covers in free lattices. Trans. Am. Math. Soc. 288, 1–42 (1985)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Freese, R., Nation, J. B.: Free and Finitely Presented Lattices. Lattice Theory: Special Topics and Applications, vol. 2, p. 2758. Birkhäuser/Springer, Cham (2016)Google Scholar
  12. 12.
    Grätzer, G.: Lattice Theory: Foundation. Birkhäuser, Basel (2011)CrossRefGoogle Scholar
  13. 13.
    McKinsey, J.C.C.: The decision problem for some classes of sentences without quantifiers. J. Symb. Logic 8, 61–76 (1943)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nation, J.B.: Finite sublattices of a free lattice. Trans. Am. Math. Soc. 269, 311–337 (1982)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nation, J.B.: On partially ordered sets embeddable in a free lattice. Algebra Univers. 18, 327–333 (1984)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nation, J. B.: Notes on Lattice Theory. http://math.hawaii.edu/~jb/math618/LTNotes.pdf
  17. 17.
    Rival, I., Wille, R.: Lattices freely generated by partially ordered sets: which can be “drawn”? J. Reine Angew. Math. 310, 56–80 (1979)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Skolem, T.: Selected Works in Logic. Edited by Jens Erik Fenstad Universitetsforlaget, Oslo, p 732 (1970)Google Scholar
  19. 19.
    Tschantz, S.T.: Infinite intervals in free lattices. Order 6, 367–388 (1990)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Whitman, P.: Free lattices. Ann. Math. 42, 325–330 (1941)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Bolyai InstituteUniversity of SzegedSzegedHungary

Personalised recommendations