Automorphisms and isotone self-maps of ordered sets with top and bottom |
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Authors: | Wei-Ping Liu Honghui Wan |
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Institution: | (1) Department of Computer Science, University of Ottawa, K1N 6N5 Ottawa, Ontario, Canada;(2) Department of Mathematics, Huazhong (Central China) University of Science and Technology, 430074 Wuhan, PR China |
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Abstract: | For an ordered setP letP
P
denote the set of all isotone self-maps on P, that is, all mapsf fromP toP such thatxy impliesf(x)f(y), and let Aut (P) the set of all automorphisms onP, that is, all bijective isotone self-maps inP
P
. We establish an inequality relating ¦P
P
¦ and ¦Aut(P)¦ in terms of the irreducibles ofP. As a straightforward corollary, we show that Rival and Rutkowski's automorphism conjecture is true for lattices. It is also true for ordered sets with top and bottom whose covering graphs are planar.Supported in part by NSERC (Grant no. A2507).Supported under an NSERC International Research Fellowship. |
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Keywords: | 06A07 06A12 05E25 20B25 |
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