Algebra universalis

, 80:11 | Cite as

Symmetric embeddings of free lattices into each other

  • Gábor CzédliEmail author
  • Gergő Gyenizse
  • Ádám Kunos
Open Access


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.


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




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


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Authors and Affiliations

  1. 1.Bolyai InstituteUniversity of SzegedSzegedHungary

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