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 Scong {{,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 Scong {{,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 Scong {{,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.
Highlights
Introduction and our resultsThere are many nice and deep results on free lattices of the variety of all lattices
With some computation based on Whitman’s condition, we can prove that P = {m1, . . . , m4}∪{m1, . . . , m4} is the cardinal sum of two 4-element chains; see Figure 1 and Lemmas 4.7 and 4.9 for an illustration and for proofs, respectively, and we prove some properties of P implying that the sublattice [P ]FL(3) generated by P in FL(3) is isomorphic to the completely free lattice CF(P ; ≤) generated by the ordered set (P ; ≤); see Corollary 4.10
We say that a subset X of a lattice freely generates if the sublattice S generated by X is a free lattice with X as the set of free generators
Summary
There are many nice and deep results on free lattices of the variety of all lattices. In addition to Theorem 1.2 on total symmetry, we have some progress in studying selfdually positioned free sublattices, which is stated as follows. Assuming that 3 ≤ κ and 3 ≤ λ are cardinal numbers, FL(λ) has a selfdually positioned sublattice isomorphic to FL(κ) if and only if the inequality max{κ, א0} ≤ max{λ, א0} holds This theorem is stronger than (1.2), the main result of Czedli [2]. M4} is the cardinal sum of two 4-element chains; see Figure 1 and Lemmas 4.7 and 4.9 for an illustration and for proofs, respectively, and (plan2) we prove some properties of P implying that the sublattice [P ]FL(3) generated by P in FL(3) is isomorphic to the completely free lattice CF(P ; ≤) generated by the ordered set (P ; ≤); see Corollary 4.10
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