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Linear extension majority cycles in height-1 orders. (English) Zbl 0708.06002
Let x, y be elements of the finite poset X, and write $$x>_ py$$ to indicate that more linear extensions of the poset have x above y than y above x. Earlier work by Fishburn showed that when the height of X is $$\geq 2$$ then $$>_ p$$ can have a cycle. It has recently been shown by Gehrlein and Fishburn that the smallest poset having a $$>_ p$$-cycle has 9 elements, and that there are exactly 5 nonisomorphic 9-element posets having $$>_ p$$-cycles. The current work addresses the case of posets of height 1. If A denotes the set of maximal non-isolated elements, and B the set of minimal non-isolated elements, it is shown that any such cycle must lie entirely within A or entirely within B. If #A$$=3$$, then there is no cycle within A. If #A$$=4$$ the smallest poset of height 1 admitting a $$>_ p$$-cycle has 15 members. If #A$$=3$$, the question is left open as to whether B can contain a cycle.
Reviewer: M.F.Janowitz

##### MSC:
 06A06 Partial orders, general
##### Keywords:
preorder; linear extensions; cycle; posets of height 1
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##### References:
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