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

14 W. DZIOBIAK, J. JEŽEK, AND R. MCKENZIE
Remark 4.1. Recall that we defined a primitive ordered set to be an ordered set
without cut-points. Thus a long block is a primitive ordered set that decomposes
internally as a cardinal sum, Q = {a}+ c I[b,c], of a one-element ordered set and a
bounded ordered set of at least three elements. In such a long block Q, the isolated
point a is called the orphan in Q, and the interval I[b 0 ,b 1 ] that makes up the rest
of Q is called the chamber in Q.
Definition 4.4. Let P be an ordered set. P has a unique internal ordinal sum
decomposition into singletons {c} (c ranging through the cut-points of P) and
primitive ordered sets. The primitive ordered subsets of P occuring in this decomposition
will be called the primitive vertical components of P. They are, of course,
convex subsets of P, and each one of them is either identical to {x : x < c} where
c is the least cutpoint of P, or of the form I(c,c ′ ) where c < c ′ are successive cutpoints,
or is identical to {x : c < x} where c is the largest cutpoint of P, or finally,
is identical with P itself if P has no cut-points.
We define critical blocks of depth n and critical intervals of depth n in P, for
any positive integer n.
Each primitive vertical component of P can be expressed in at least one way as
a tower. By a critical block of depth 1 in P, we mean a long block that occurs as
a block in some tower expression for some primitive vertical component of P. By
a critical interval of depth 1 in P we mean the chamber of some critical block of
depth 1 in P.
Now suppose that a 0 ,...,a n−1 is an n-element anti-chain in P and
b 0 < b 1 < ··· < b n−1 < c n−1 < c n−2 < ··· < c 1 < c 0
is a chain in P such that the following hold.
• a i isincomparabletoallelementsofI[b i ,c i ]for0 ≤ i < nandb i < a i+1 < c i
for 0 ≤ i < n−1.
• Where Q i = {a i }∪I[b i ,c i ], Q 0 is a critical block of depth 1 in P and for
all i < n−1, Q i+1 is a critical block of depth 1 in I[b i ,c i ].
In this situation, we call Q n−1 a critical block of depth n in P, and its chamber,
I[b n−1 ,c n−1 ], a critical interval of depth n in P.
The critical intervals and critical blocks of depth n in P are precisely those
intervals and blocks for which the auxiliary elements exist as above.
Notice that in the above situation the elements a 0 ,...,a n−1 form an n-element
anti-chain of elements incomparable to all elements of I[b n−1 ,c n−1 ]. Thus
Lemma 4.5. If I[b,c] is a critical interval of depth n in P then w(P) ≥ n +
w(I[b,c]).
Definition 4.6. Let P be a poset. An interval I[b 0 ,b 1 ] ⊆ P is undivided (in P) iff
b 0 < b 1 and for all x ∈ P \I[b 0 ,b 1 ], and all b 0 < y < b 1 , if x < y then x < b 0 , and
if y < x then b 1 < x. Otherwise, the interval is said to be divided (in P).
Lemma 4.7. Suppose that Q = {a} + c I[b 0 ,b 1 ] is a long block, and c 0 < c 1 are
successive cut-points of I[b 0 ,b 1 ]. Then the interval I[c 0 ,c 1 ] is undivided in Q.