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

20 W. DZIOBIAK, J. JEŽEK, AND R. MCKENZIE
element of depth d 0 in Y 0 below x 0 ; and likewise mapping, for each 0 ≤ j < l 1 , the
two elements of height j in Y 1 ′ to the two elements of height j in Y 1 , making sure,
if ¯c 1 = 〈l 1 ,d 1 ,1,x 1 〉 and ¯c ′ 1 = 〈l ′ 1,d 1 ,1,x 1 〉, to map the element of height d 1 in Y 1
′
above x ′ 1 = x 1 to the element of height d 1 in Y 1 above x 1 .
The detailed verification that this works is left to the reader. •
Remark 5.1. An inspection of the proof of Theorem 5.6 reveals the following
refined version of the conclusion of the theorem:
(1) If Y 0 = ∅ = Y 1 , then S(P 0 ,...,P n−1 ) ∂ is isomorphic to a 0,1-sublattice of
S ′ (P ′ 0,...,P ′ n−1) ∂ .
(2) If Y 0 = ∅, then S(P 0 ,...,P n−1 ) ∂ is isomorphic to a 0-sublattice of
S ′ (P ′ 0,...,P ′ n−1) ∂ .
(3) If Y 1 = ∅, then S(P 0 ,...,P n−1 ) ∂ is isomorphic to a 1-sublattice of
S ′ (P ′ 0,...,P ′ n−1) ∂ .
These statements will be used in the proof of Theorem 7.1.
The next theorem justifies all the definitions above.
Theorem 5.7. Let N ≥ 3. Suppose that P is a primitive ordered set belonging to
P N . Then P is nicely (M)-structured, for M = 5·(18N) 6N+4 . In fact, P admits
a nice (M)-system S = (X,Y 0 ,Y 1 ;b 0 ,...,b n−1 ;c 0 ,...,c n−1 ) such that
(i) if h ⋄ (P) < N then Y 0 = ∅ = Y 1 ; and
(ii) if D k + ′ o B 2,3 ≰ 0 P ∂ for every k ≥ N then either Y 0 = ∅, or P = Y 0 is a
ladder and hence X ∪Y 1 = ∅ and n = 0.
This theorem is established with the next proposition, which will be proved by
induction.
Proposition 5.8. Let P be a primitive ordered set in P N , N ≥ 3. For each k,
1 ≤ k < w(P), P admits an ((18N) k )-system S k such that all intervals of S k
that are divided in P are critical intervals of depth k in P (and thus w(S k ) ≤
w(P)−k). Moreover, the system S k can be chosen so that the assertions expressed
in statements (i) and (ii) of Theorem 5.7 are true of it.
The base step in the inductive argument is the case k = 1, established in the
next lemma.
Lemma 5.9. Let P be a primitive ordered set in P N , N ≥ 3. Then P admits an
(M)-system S 1 , M ≤ 3(3N + 7) ≤ 18N, such that all intervals in S 1 are critical
intervals of depth 1 in P and the assertions expressed in statements (i) and (ii) of
Theorem 5.7 are true of S 1 .
Proof. According to Lemma 4.9, P admits a fitting tower
Z 0 0,...,Z 0 r−1,M 0 ,...,M s−1 ,Z 1 0,Z 1 1,...,Z 1 t−1
where r+s+t = h ⋄ (P), Z 0 0,...,Z 0 r−1 and Z 1 0,...,Z 1 t−1 are ladders and s ≤ N +3.
(Any of r,s,t may be 0.) Moreover, by Remark 4.2, we can assume that r = 0 = s
if h ⋄ (P) < N; and if D k + ′ o B 2,3 ≰ 0 P ∂ for every k ≥ N then we have r ≤ N −1.