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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References
B. Birkhoff (1967)Lattice Theory (3rd ed.), AMS, Providence.
D. Duffus, V. Rödl, B. Sands, and R. Woodrow (1992) Enumeration of order preserving maps,Order 9, 15–29.
W.-P. Liu, I. Rival, and N. Zaguia (1991) Automorphisms, isotone self-maps and cycle-free ordered sets, manuscript.
H. J. Prömel (1987) Counting unlabelled structures,J. Combin. Theory A 44, 83–93.
I. Rival and A. Rutkowski (1990) Does almost every isotone self-maps have a fixed point?Colloq. Math. Janos Bolyai, to appear.
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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