Abstract
Let P be a finite poset covered by three nonempty disjoint chains T 1, T 2, and T 3. Suppose that p and q are different members of P. Also, P has the property that if p and q are in different chains and p < q, then P = above … ∪ below …. D.E. Daykin and J.W. Daykin (1985) made the conjecture: “There is a partition P = R 1 ∪ R 2 ∪ … ∪ R n such that R 1 < R 2 < … < R n . For each integer i, 1 ⩽ i ⩽ n, either R i and T j are disjoint for some j in 1,2, 3, or if p and q are members of R i , then we have the following property: If p and q are in different chains, then p and q are incomparable.” In this paper, we give the complete structural details of this conjecture and prove it.
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