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

18 W. DZIOBIAK, J. JEŽEK, AND R. MCKENZIE
In this situation, we also say that S is a skeleton for S.
Where M is a positive integer, a skeleton or basic skeleton for S, as above, is an
(M)-skeleton for S, or a basic (M)-skeleton for S iff |X ∪ {b i : i < n}∪{c i : i <
n}| ≤ M.
Definition 5.2. Suppose that S is an ordered set,
S = X ∪Y 0 ∪Y 1 ∪{b i : i < n}∪{c i : i < n},
and S = (X,Y 0 ,Y 1 ;b 0 ,...,b n−1 ;c 0 ,...,c n−1 ) is a skeleton for S. We define the
character of the skeleton S. Suppose that Y i is an l i -ladder (i ∈ {0,1}). (Note that
l i = 0 signifies that Y i = ∅.) Put ¯c 0 = 〈l 0 〉 if Y 0 < x for all x ∈ S \ Y 0 . In the
contrary case, put ¯c 0 = 〈l 0 ,d 0 ,ε 0 ,x 0 〉 where d 0 , ε 0 , x 0 are defined as follows: x 0 is
the unique element x ∈ S \ Y 0 such that Y 0 < x fails. We have 0 ≤ d 0 < l 0 and
x 0 fails to be above both elements of depth d 0 in Y 0 , while it is above all elements
of Y 0 of depth greater than d 0 . ε 0 is the number of elements at depth d 0 in Y 0
that are below x 0 (either 0 or 1). We define ¯c 1 dually (so to speak). That is, we
put ¯c 1 = 〈l 1 〉 if x < Y 1 holds for all x ∈ S \ Y 1 . In the contrary case, we put
¯c 1 = 〈l 1 ,d 1 ,ε 1 ,x 1 〉 where d 1 , ε 1 , x 1 are defined as follows: x 1 is the unique element
x ∈ S\Y 1 such that x < Y 1 fails. We have 0 ≤ d 1 < l 1 and x 1 fails to be below both
elements of height d 1 in Y 1 , while it is below all elements of Y 1 of height greater
than d 1 . ε 1 is the number of elements at height d 1 in Y 1 that are above x 1 (either
0 or 1).
The character of S is the pair (¯c 0 ,¯c 1 ).
Definition 5.3. Suppose that S = (X,Y 0 ,Y 1 ;b 0 ,...,b n−1 ;c 0 ,...,c n−1 ) is a skeletonforanorderedsetS,
andS ′ = (X,Y 0,Y ′
1;b ′ 0 ,...,b n−1 ;c 0 ,...,c n−1 )isaskeleton
for an ordered set S ′ having the same underlying basic skeleton (X;b 0 ,...,b n−1 ;
c 0 ,...,c n−1 ). Let (¯c 0 ,¯c 1 ), (¯c ′ 0,¯c ′ 1) be the characters of S and S ′ , respectively. We
write (¯c 0 ,¯c 1 ) ≤ (¯c ′ 0,¯c ′ 1) to mean that for i = 0 and i = 1:
• ¯c i = 〈l i 〉 iff ¯c ′ i = 〈l′ i 〉 and if ¯c i = 〈l i 〉 then l i ≤ l ′ i ;
• if ¯c i = 〈l i ,d i ,ε i ,x i 〉 and ¯c ′ i = 〈l′ i ,d′ i ,ε′ i ,x′ i 〉 then x i = x ′ i , ε i = ε ′ i , d i ≤ d ′ i
and l i ≤ l ′ i .
Definition 5.4. Let P be a finite ordered set, and M be a positive integer.
By a basic (M)-system for P we mean an ordered subset S ⊆ P together with
a basic (M)-skeleton for S, S = (X;b 0 ,...,b n−1 ;c 0 ,...,c n−1 ), such that P =
S ∪ ⋃ 0≤i