Abstract
For an ordered setP letP P denote the set of all isotone self-maps on P, that is, all mapsf fromP toP such thatx≥y 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.
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Communicated by N. Zaguia
Supported in part by NSERC (Grant no. A2507).
Supported under an NSERC International Research Fellowship.
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Liu, WP., Wan, H. Automorphisms and isotone self-maps of ordered sets with top and bottom. Order 10, 105–110 (1993). https://doi.org/10.1007/BF01111294
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DOI: https://doi.org/10.1007/BF01111294