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

AVOIDABLE STRUCTURES, II 19
Definition 5.5. An (M)-system S = (X,Y 0 ,Y 1 ;b 0 ,...,b n−1 ;c 0 ,...,c n−1 ) or basic
(M)-system S = (X;b 0 ,...,b n−1 ;c 0 ,...,c n−1 ) for P is said to be nice if none of the
intervals I[b i ,c i ], 0 ≤ i < n, is divided in P. We say that P is nicely (M)-structured
iff P admits a nice (M)-system.
Suppose that P admits the nice (M)-system
where
S = (X,Y 0 ,Y 1 ;b 0 ,...,b n−1 ;c 0 ,...,c n−1 ),
S = X ∪Y 0 ∪Y 1 ∪{b 0 ,...,b n−1 }∪{c 0 ,...,c n−1 }
is an ordered subset of P. Then P arises through the following construction. We
have the (M)-skeleton S for S, and the pairwise disjoint ordered sets (ordered
subsets of P) P i = I(b i ,c i ) P , 0 ≤ i < n, which are all disjoint from S. The ordered
set
S(P 0 ,...,P n−1 )
(which is actually identical to P) is then defined as follows: The universe of
S(P 0 ,...,P n−1 ) is S ∪ ⋃ i P i. The order is defined to be the transitive closure
of the union of the order on S and the orders on each P i and the pairs b i < x and
x < c i for all x ∈ P i and all 0 ≤ i < n. In this ordered set S(P 0 ,...,P n−1 ) (= P),
we have that each P i = I(b i ,c i ) P , and each interval I[b i ,c i ] is undivided in P.
The utility of these ideas is manifested in this theorem.
Theorem 5.6. Suppose that S = (X,Y 0 ,Y 1 ;b 0 ,...,b n−1 ;c 0 ,...,c n−1 ) and S ′ =
(X,Y 0,Y ′
1;b ′ 0 ,...,b n−1 ;c 0 ,...,c n−1 ) are two (M)-skeletons with the same underlying
basic (M)-skeleton. Suppose that P i (0 ≤ i < n) are pairwise disjoint ordered
sets disjoint from the universe of S and P i ′ (0 ≤ i < n) are pairwise disjoint ordered
sets disjoint from the universe of S ′ . If the characters (¯c 0 ,¯c 1 ), (¯c ′ 0,¯c ′ 1) of S and
S ′ satisfy (¯c 0 ,¯c 1 ) ≤ (¯c ′ 0,¯c ′ 1) and for all 0 ≤ i < n we have that Pi ∂ ≤ P i∂ ′ , then
S(P 0 ,...,P n−1 ) ∂ ≤ S ′ (P 0,...,P ′ n−1) ′ ∂ .
Proof. Under all the given assumptions, we need to find a monotone mapping of
some convex subset of P ′ = S ′ (P 0,...,P ′ n−1) ′ onto P = S(P 0 ,...,P n−1 ).
We do have for each i, a monotone mapping of a convex subset Q i ⊆ P i ′ onto P i
(by Birkhoff duality). We can extend this (in several possible ways) to a monotone
mapping f i of I[b i ,c i ] P ′ onto I[b i ,c i ] P . Setting f = id X ∪ ⋃ i f i we have that f
is a monotone map of P ′ \ (Y 0 ′ ∪ Y 1) ′ onto P \ (Y 0 ∪ Y 1 ). This is true because of
the way in which the order on P ′ \ (Y 0 ′ ∪ Y 1) ′ (respectively, on P \ (Y 0 ∪ Y 1 )) is
the minimum extension of the order on X ∪{b 0 ,...,b n−1 }∪{c 0 ,...,c n−1 } joined
to the disjoint union of the orders on the individual I[b i ,c i ] P ′ (respectively, on the
individual I[b i ,c i ] P ).
It remains to extend the monotone mapping f to map a convex subset of P ′
onto P. We do this by adding to the domain of f the top l 0 levels of Y 0 ′ and the
bottom l 1 levels of Y 1 ′ and mapping, for each 0 ≤ j < l 0 , the two elements of depth
j in Y 0 ′ to the two elements of depth j in Y 0 , making sure, if ¯c 0 = 〈l 0 ,d 0 ,1,x 0 〉
and ¯c ′ 0 = 〈l ′ 0,d 0 ,1,x 0 〉, to map the element of depth d 0 in Y 0 ′ below x ′ 0 = x 0 to the