Let n >= 3. The poset dn(1) consists of the 2n-2 elements named as follows, and which possess only the indicated covering relations: w1 -> w2 -> ... -> wn-2 -> x,y -> zn-2 -> ... -> z2 -> z1 . For example, d3(1) is the "diamond" poset consisting of 4 elements. The poset dn(1)- is defined to be the poset obtained by removing the maximal element z1 from dn(1). (The main aspect of the d-complete condition for a poset P is that P does not contain any convex subset which does not quite form a dk(1) poset and which cannot be extended to form an interval which is a dk(1) poset.) Let P be a poset. Let k >= 3. Let S be a convex set in P. We say that S is a dk--convex set if it is isomorphic to the poset dk(1)-. Remark: If an element z covers exactly the maximal elements of S, then the union of S with {z} is isomorphic to dk(1). Here we say that z completes S.
A poset P is d-complete if for every dk--convex set S, k >= 3, there is an element which covers exactly the maximal elements of S and which does not cover the maximal elements of another dk--convex set.
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