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.
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Dedicated to Ralph Freese and J. B. Nation on their seventieth birthdays.
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This research was supported by the National Research, Development and Innovation Fund of Hungary under the KH 126581 funding scheme.
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Czédli, G., Gyenizse, G. & Kunos, Á. Symmetric embeddings of free lattices into each other. Algebra Univers. 80, 11 (2019). https://doi.org/10.1007/s00012-019-0583-7
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DOI: https://doi.org/10.1007/s00012-019-0583-7