Partitions of a finite three-complete poset

AbstractLet P be a finite poset covered by three nonempty disjoint chains T1, T2, and T3. 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 = R1 ∪ R2 ∪ … ∪ Rn such that R1 < R2 < … < Rn. For each integer i, 1 ⩽ i ⩽ n, either Ri and Tj are disjoint for some j in 1,2, 3, or if p and q are members of Ri, 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