Symmetric embeddings of free lattices into each other
Gábor Czédli, Gergő Gyenizse, Ádám Kunos · Algebra Universalis · 2019
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.