Linear extension majority cycles on partial orders |
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Authors: | William V Gehrlein Peter C Fishburn |
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Institution: | (1) Department of Business Administration, University of Delaware, 19716 Newark, DE, USA;(2) AT&T Bell Laboratories, 600 Mountain Avenue, 07974 Murray Hill, NJ, USA |
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Abstract: | Let P define a partial order on a set X of cardinalityn. A linear extensionL ofP is a linear order withP G L, and
is the set of all linear extensions ofP.
denotes that subset of
withxLy forx, y X. A linear extension majority (LEM) relationM onX is defined byxMy if
. Similarly,M is defined byxM y if
. An LEM cycle exists if there arex, y, z X withxMyMzMx, and an LEM quasi-cycle exists ifxM yM zM x and the equality part of the definition ofM holds for exactly one pair in the triple. The study shows that no semiorders have LEM cycles or LEM quasi-cycles, and that every interval order has a maximal element under theM relation. LEM cycles and LEM quasi-cycles are also considered for partial orders with specific structures. Simulation is used to determine the relative likelihood with which LEM cycles and LEM quasi-cycles are observed when connected partial orders are generated at random by a specific procedure.Dr. Gehrlein's research was supported through a fellowship from the Center for Advanced Study of the University of Delaware. |
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