Symmetric embeddings of free lattices into each other

TL;DR

This paper constructs highly symmetric sublattices within free lattices, especially embedding countably infinite free lattices into three-generator free lattices in a totally symmetric way, and characterizes when such embeddings are possible.

Contribution

It introduces the concept of totally symmetric embeddings of free lattices and determines all such embeddings between free lattices of various infinite cardinalities.

Findings

Constructed a sublattice of FL(3) fixed by all automorphisms, embedding FL(ω) symmetrically.

Characterized all pairs (κ, λ) allowing totally symmetric embeddings of FL(κ) into FL(λ).

Developed a computer program to verify calculations related to free lattice word problems.

Abstract

By a 1941 result of Ph. M. Whitman, the free lattice FL(3) on three generators includes a sublattice $S$ that is isomorphic to the lattice FL($\omega$)=FL($\aleph_0$) generated freely by denumerably many elements. The first author has recently "symmetrized" this classical result by constructing a sublattice $S\cong$ FL($\omega)$ of FL(3) such that $S$ is SELFDUALLY POSITIONED in FL(3) in the sense that it is invariant under the natural dual automorphism of 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 $S\cong$ FL$(\omega)$ of FL(3) such that every element of $S$ is fixed by all automorphisms of FL(3). That is, in our terminology, we embed FL$(\omega)$ into FL(3) in a TOTALLY SYMMETRIC way. Our main result determines all pairs $(\kappa,\lambda)$ of cardinals greater than 2 such that FL$(\kappa)$ is embeddable into FL$(\lambda)$ in a totally symmetric way. Also, we relax the stipulations on $S\cong$FL$\kappa$ by requiring only that $S$ is closed with respect to the automorphisms of 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.