The Central Decomposition of FD01(n)

The paper presents a method of composing finite distributive lattices from smaller pieces and applies this to construct the finitely generated free distributive lattices from appropriate Boolean parts.


Introduction
The free distributive lattice F D 01 (3) on three generators as drawn in Fig. 1 can be viewed as a sort of composition of four Boolean lattices layered on top of each other, with the three generators a, b, c serving as additional merging points. Another way of seeing this is via the 'central' elements 0, p, q, r, 1, where the intervals [0, p], [p, q], [q, r], [r, 1] form the respective Boolean lattices.
In this paper we will show, that this behaviour can be found in all the finitely generated free distributive lattices. Moreover we will give a nonrecursive construction of these lattices from their Boolean lattice building blocks.
The original hope that this might provide a better way to compute their cardinalities did not materialize. As in the known recursive attempts (see e.g. [2,11]) also this approach requires an addition of interval sizes which for larger values of n goes beyond the capacities of current computers.

Lattices
In the following all lattices L are finite distributive lattices with a 0-element 0 L and a 1element 1 L . By B n we denote the Boolean lattice with elements 0,...,2 n − 1 and binary join and meet. In particular 0 B n = 0 and 1 B n = 2 n − 1, and the atoms of B n are 1, 2, ..., 2 n−1 .
For elements a and b of a lattice L we denote by (a] the principal ideal (or 'downset') {x|x ∈ L, x ≤ a}, by [a) the principal filter (or 'upset') {x|x ∈ L, x ≥ a} and by [a, b] the interval {x|x ∈ L, a ≤ x ≤ b}. We start with an easy observation belonging to the folklore of distributive lattices (see e.g. [1,6]): Theorem 1 Let L be a finite distributive lattice and a ∈ L. Then What is less known is the following reverse construction, which has its origin in the general theory of 'triple sums' originally developed in [4,5] and later extended in [7][8][9][10].  Proof (i) is obvious. For (ii) let a = (1 L , 0 M ). Then a ∈ L ⊗ φ M and the mappings x → (x, 0 M ) and y → (1 L , y) are clearly isomorphisms from L to (a] and M to [a).
where * denotes the relative pseudocomplement, i.e. a * x = {z|z ∈ L, z ∧ a ≤ x}. Then it is well known from the theory of pseudocomplented lattices that φ has the required properties.
Let us note that the theorem above could have also formulated using the notion of split exact sequences (see [9,10]).
There are three well known special cases: Another use of the skew square can be seen in the following easy observation: Theorem 3 Let B n be the Boolean lattice of order 2 n , and let C 3 be the three element chain.
Proof This is obvious for n = 1, the rest follows by an easy induction argument, enumerating pairs of pairs in two different ways.
Another interesting observation concerning the 'skew square' of a composition is: Proof Obviously ψ and χ are 1-meet-preserving. Now by definition )} and by the definition of ψ and χ these conditions coincide.
So far we have only considered the composition of pairs of distributive lattices. Now if L ⊗ φ M and M ⊗ ψ N are two such compositions, then these give rise to two combinations, where φ * and ψ * are the natural extensions of φ and ψ defined by Obviously both compositions amount to the same set, namely {(l, m, n)|l ∈ L, m ∈ M, n ∈ N, m ≤ φ(l), n ≤ ψ(m)}. Therefore it makes sense to introduce the notion of a triple composition L ⊗ φ M ⊗ ψ N , and more generally that of an n-fold composition And as such an (n+1)-fold composition we will construct F D 01 (n). However, before turning to the general case we describe the construction of F D 01 (3) as a quadruple where φ 0 and φ 2 are the 0-mappings and φ 1 : That this really gives F D 01 (3), can be seen from its diagram in the canonical numbering as in Fig. 2 and the expression of the element numbers as 4-tuples as in Table 1, where the correspondence is given by For the general case of n ∈ N this suggests to use the Boolean lattices B ( n k ) corresponding to the binomial coefficients n k for k = 0, ..., n as building blocks.  The key to the proof is the following generalization of Lemma 1: .., L n be distributive lattices and for 0 ≤ i < n let φ i : L i → L i+1 be 1-meet-preserving maps. Then where the mappings ψ 1 : ) for all x i ∈ L i , y i−1 ∈ L i−1 , 0 < i < n ψ n ((y n−1 , x n )) = x n ∧ φ n−1 (y n−1 ) for all x n ∈ L n , y n−1 ∈ L n−1 .

Proof of Theorem 4 The result is immediate for
where φ 0 is the 0-map. Now assume that the result holds for n ≥ 1. As in Example (iii) on page 3 we have that F D 01 (n + 1) ∼ = F D 01 (n) F D 01 (n). By the induction hypothesis and Lemma 2 we get F D 01 (n Now the fact that B i × B j ∼ = B i+j for all i, j ∈ N and the addition rules for the binomial coefficients show that the statement of the theorem holds also for n + 1.
In this proof the crucial mappings ψ 0 , ..., ψ n are defined recursively. It is, however, possible to give a direct definition. We defer this to the next section.

Posets
An element x of a lattice L is called meet irreducible, if it cannot be expressed as a meet of greater elements, i.e. x = y ∧ z implies x = z or x = y. In particular, 1 L is not meet irreducible. The poset of meet irreducible elements of L is denoted by M (L).

Fig. 3 Sum of two antichains
A subset I of a poset P is called an ideal, if it it "downward closed", i.e. p ∈ I and q ≤ p implies q ∈ I . In particular, ∅ and P are ideals of P . By I (P ) we denote the set (lattice) of ideals of P .
We start this section with the poset counterpart of the triple construction for lattices: Theorem 5 Let P , Q be finite posets and α : Q → I (P ) be an order preserving mapping. Then the set P ⊕ α Q = P∪ Q equipped with the relation ≤ defined by Proof Clearly ≤ is reflexive and antisymmetric. To show that it is transitive too, it suffices to consider three elements x, y, z with x ≤ y and y ≤ z and the two nontrivial cases (i) x ∈ P , y ∈ Q, z ∈ Q and (ii) x ∈ P , y ∈ P , z ∈ Q. Now for (i) transitivity comes from the fact that α is order preserving, and for (ii) from the fact that α(z) is an ideal.
Proof Clearly φ(X) is an ideal of Q for every X ∈ I (P ), so φ is a mapping. It is 1-meetpreserving as well. We now observe that for any (X, Y ) ∈ I (P ) ⊗ φ I (Q) the set X∪Y is an ideal of P ⊕ α Q. In fact let y ∈ X∪Y and x ≤ y. In order to show that x ∈ X∪Y too, is suffices to consider the case x ∈ P and y ∈ Q. But then we have x ∈ α(y) and hence x ∈ X. This implies we can define a mapping χ : I (P ) ⊗ φ I (Q) → I (P ⊕ α Q) by χ(X, Y ) = X∪Y . Its inverse is given by Z → (Z ∩ P )∪(Z ∩ Q) and since both are order preserving they are lattice isomorphisms too.
As already indicated, we will apply this result to obtain a nonrecursive definition of the composition mappings φ k of Theorem 2. In order to facilitate this we introduce some notation: For n ∈ N let be the set of all k-element subsets of P n . Then P n can be decomposed into antichain layers as P n = S n,0∪ S n,1∪ ...∪ S n,n .
With the mappings α k : S n,k+1 → I (S n,k ) defined by we can even generalize the composition to where we tacitly extend the poset triple sum to an n-fold sum.
Repeatedly applying Theorem 4 we arrive at: Proof It is well known that F D 01 (n) ∼ = I (P n ) (see e.g. [3]). Moreover, as S n,k is an antichain, it is clear that I (S n,k ) = P(S n,k ). So the only thing that remains to be shown, is that the formula given for φ n,k is equivalent to the one obtained from Theorem 7 -but that is obvious too.
To illustrate the definition of φ n,k we list some values for n = 4 in Table 2, where we restrict ourselves to list the mapping values for the topmost elements, i.e. the 1-element and the dual atoms:

Computations
Even though Theorem 8 gives a direct, nonrecursive construction, its application to determine the cardinalities for larger values of n fails with respect to the slowness of computing the 'downsets' of the partial compositions.
To see this in some more detail let us recall that the definition of the composition L⊗ φ M implies that for any (x, y) ∈ L ⊗ φ M we have Applying this repeatedly to the formula of Thereom 2 we end up with ...  We have carried out a computer calculation of these sequences up to n = 6. Tables 3  and 4 list the values of the mappings φ n,k and the respective c-values for n = 4. Note that in Table 3 the columns contain the nonzero function values and in Table 4 the three columns for each of the Boolean lattices contain representative elements, their c-value and the number of elements with the same value.
Concluding remarks It might be worthwhile to try to use some insight into the known structure of the Boolean lattices L 0 ,..., L n to speed up the computation.
Another speedup approach would be the use of the induced action of the symmetric group S n on the lattices L 1 , ...L n−1 , as this was successfully done in [11].