Avoidable structures, II: finite distributive lattices and nicely ...

AVOIDABLE STRUCTURES, II 29
Claim 4: There are only finitely many n for which Q n is a ladder.
Continuing with Case 1.2, we next observe that since P n ∈ P 2 N and so P∂ n ≱ 2 J 1 k
for all k ≥ N, it follows that Q ∂ n ≱ 0 D k + ′ o B 2,3 for every k ≥ N. Thus by
Claim 4 and Theorem 5.7 (especially statement (ii) of the theorem), Q n is nicely
(M)-structured, M = 5·(18N) 6N+4 , in fact admits a nice (M)-system
S n = (X,∅,Y 1 ;b 0 ,...,b kn−1;c 0 ,...,c kn−1).
As we did above for Case 1.1, we now work through the steps of our argument
for Theorem 6.1.
By cutting down (and replacing some P n by isomorphic ordered sets), we can
assume that for all n, the basic skeletons
are the same sets, i.e.,
S ′ n = (X;b 0 ,...,b kn−1;c 0 ,...,c kn−1)
S ′ n = S ′ = (X;b 0 ,...,b k−1 ;c 0 ,...,c k−1 );
and have the same induced order from Q n . Thus we can write
P n = S n (Q 0 n,...,Q k−1
n )+ o 1.
It is true, also in Case 1.2, that for 0 ≤ r < k, I Pn [b r ,c r ] = Q r n ∪ {b r ,c r } is a
bounded convex subset of P n , implying that there is a monotone map of P n onto
this interval. Thus I Pn [b r ,c r ] = I Qn [b r ,c r ] ∂ < 2 L n .
Now the proof goes just as in Case 1.1, except in the very last step, where the
application of Remark 5.1 following Theorem 5.6 gives that where A n = Q ∂ n (and
L n
∼ = An + o 1) there is i < j with A i ≤ 0 A j . Clearly, this implies that L i ≤ 2 L j .
This contradiction finishes our proof of this theorem in Case 1.
Case 2: i = 1. The directimplication follows from thefact that the setconsisting
of B 2,3 + ′ oD n (n ≥ 1) and J n + o 1 (n ≥ 0) is an infinite anti-chain with respect to
≤ 1 . The converse follows from the case i = 2 just proved and the observations that
for any A,B ∈ D, A ≤ 1 B iff 1+ o A ≤ 2 1+ o B, and A ∈ A 1 N iff 1+ o A ∈ A 2 N .
Finally, the result for the pre-order ≤ 0 is the dual of the result for Case 2, so we
omit the proof of it.
•
Recall that I 0 is the lattice 1+ o B 2,3 + o 1 and I 3 is the eight-element Boolean
lattice.
Theorem 7.2. The following are true for a finite distributive lattice L:
(1) L is avoidable in 〈D,≤ 2 〉 if and only if it has a 0,1-sublattice isomorphic
to either 1 or D 1 or D 1 + o 1 or 1+ o D 1 or I 0 .
(2) L is avoidable in 〈D,≤ 1 〉 if and only if it has a 1-sublattice isomorphic to
either D 1 or I 0 or I 3 + o 1.
(3) L is avoidable in 〈D,≤ 0 〉 if and only if it has a 0-sublattice isomorphic to
either D 1 or I 0 or 1+ o I 3 .