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Partitions of a finite three-complete poset

✍ Scribed by Shiojenn Tseng; Muh-Chyi Horng

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
549 KB
Volume
148
Category
Article
ISSN
0012-365X

No coin nor oath required. For personal study only.

✦ Synopsis

Let P be a finite poset covered by three nonempty disjoint chains 7"1, 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{p} u below{q}. D.E. Daykin and J.W. Daykin (1985) made the conjecture: "There is a partition P = R~ u R2 w ... w Rn such that R1 < R2 < ... < R,. 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.

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## Abstract Let __V__~n~(q) denote a vector space of dimension __n__ over the field with __q__ elements. A set ${\cal P}$ of subspaces of __V__~n~(q) is a __partition__ of __V__~n~(q) if every nonzero element of __V__~n~(q) is contained in exactly one element of ${\cal P}$. Suppose there exists a p