# Symmetric embeddings of free lattices into each other

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## 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.Czédli, G.: On the word problem of lattices with the help of graphs. Period. Math. Hung.
**23**, 49–58 (1991)MathSciNetCrossRefGoogle Scholar - 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.Dean, R.A.: Completely free lattices generated by partially ordered sets. Trans. Am. Math. Soc.
**83**, 238–249 (1956)MathSciNetCrossRefGoogle Scholar - 4.Dean, R.A.: Free lattices generated by partially ordered sets and preserving bounds. Can. J. Math.
**16**, 136–148 (1964)MathSciNetCrossRefGoogle Scholar - 5.Dilworth, R.P.: Lattices with unique complements. Trans. Am. Math. Soc.
**57**, 123–154 (1945)MathSciNetCrossRefGoogle Scholar - 6.Evans, T.: The word problem for abstract algebras. Lond. Math. Soc.
**26**, 64–71 (1951)MathSciNetCrossRefGoogle Scholar - 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.Freese, R.: Free lattice algorithms. Order
**3**, 331–344 (1987)MathSciNetCrossRefGoogle Scholar - 9.Freese, R., Ježek, J., Nation, J. B.: Free Lattices. Mathematical Surveys and Monographs, vol. 42. American Mathematical Society, Providence (1995)Google Scholar
- 10.Freese, R., Nation, J.B.: Covers in free lattices. Trans. Am. Math. Soc.
**288**, 1–42 (1985)MathSciNetCrossRefGoogle Scholar - 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.Grätzer, G.: Lattice Theory: Foundation. Birkhäuser, Basel (2011)CrossRefGoogle Scholar
- 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.Nation, J.B.: Finite sublattices of a free lattice. Trans. Am. Math. Soc.
**269**, 311–337 (1982)MathSciNetCrossRefGoogle Scholar - 15.Nation, J.B.: On partially ordered sets embeddable in a free lattice. Algebra Univers.
**18**, 327–333 (1984)MathSciNetCrossRefGoogle Scholar - 16.Nation, J. B.: Notes on Lattice Theory. http://math.hawaii.edu/~jb/math618/LTNotes.pdf
- 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.Skolem, T.: Selected Works in Logic. Edited by Jens Erik Fenstad Universitetsforlaget, Oslo, p 732 (1970)Google Scholar
- 19.Tschantz, S.T.: Infinite intervals in free lattices. Order
**6**, 367–388 (1990)MathSciNetCrossRefGoogle Scholar - 20.Whitman, P.: Free lattices. Ann. Math.
**42**, 325–330 (1941)MathSciNetCrossRefGoogle Scholar

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