Let n >= 3. The poset dn(1) is defined to be the poset that consists of 2n-2 elements such that: Except for two elements which are incomparable, every element is comparable to every other element, and each of the two incomparable elements is less than n-2 elements. Hence the two incomparable elements are each greater than n-2 elements. Hasse diagrams: The Hasse diagram of d3(1) is a diamond. For n >= 4, the Hasse diagram of dn(1) is obtained by drawing a chain of n-3 elements above the top element of a diamond and a chain of n-3 elements below the bottom element of that diamond. The poset dn(1)- is defined to be the poset obtained by removing the maximal element from dn(1). Remark: The foremost aspect of the d-complete condition for a poset is that there is no convex subset that forms a dk(1)- poset and that cannot be extended to form an interval that is a dk(1) poset. Let k >= 3. Let S be a convex set in a poset. We say that S is a dk--convex set if it is isomorphic to dk(1)-. Fact: If an element z covers exactly the maximal elements of such a subset S, then the union of S with {z} is isomorphic to dk(1). Here we say that z completes S.
A poset is d-complete if for all k >= 3 and every dk--convex set S, there is an element that covers exactly the maximal elements of S and that does not cover the maximal elements of any other dk--convex set.
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