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The endomorphism spectrum of an ordered set

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Abstract

Theendomorphism spectrum of an ordered setP, spec(P)={|f(P)|:f ∈ End(P)} andspectrum number, sp(P)=max(spec(P)\{|P|}) are introduced. It is shown that |P|>(1/2)n(n − 1)n − 1 implies spec(P) = {1, 2, ...,n} and that if a projective plane of ordern exists, then there is an ordered setP of size 2n 2+2n+2 with spec(P)={1, 2, ..., 2n+2, 2n+4}. Lettingh(n)=max{|P|: sp(P)⩽n}, it follows thatc 1 n 2⩽h(n)⩽c 2 n n+1 for somec 1 andc 2. The lower bound disproves the conjecture thath(n)⩽2n. It is shown that if |P| − 1 ∈ spec(P) thenP has a retract of size |P| − 1 but that for all κ there is a bipartite ordered set with spec(P) = {|P| − 2, |P| − 4, ...} which has no proper retract of size⩾|P| − κ. The case of reflexive graphs is also treated.

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References

  1. Duffus, D., Rödl, V., Sands, B., and Woodrow, R. (1992) Enumeration of order preserving maps,Order 9, 15–29.

    Google Scholar 

  2. Duffus, D., Rival, I., and Simonovits, M. (1980) Spanning retracts of a partially ordered set,Discrete Math. 32, 1–7.

    Google Scholar 

  3. Hedrlin, Z. and Pultr, A. (1966) On rigid undirected graphs,Canad. J. Math. 18, 1237–1242.

    Google Scholar 

  4. Hell, P. and Nešetřil, J. (1978) Homomorphisms of graphs and of their orientations,Monatsh. für Math. 85, 39–48.

    Google Scholar 

  5. Hell, P. and Nešetřil, J. (1991) Images of rigid digraphs,Europ. J. Combinatorics 12, 33–42.

    Google Scholar 

  6. Hughes, D. R. (1960) On homomorphisms of projective planes,Proc. Symp. Appl. Math. 10, 45–52.

    Google Scholar 

  7. Hughes, D. R. and Piper, F. C. (1973)Projective Planes, Springer-Verlag, New York.

    Google Scholar 

  8. Jawhari, E., Misane, D., and Pouzet, M. (1986) Retracts: Graphs and ordered sets from the metric point of view,Combinatorics and Ordered Sets, Contemporary Mathematics57, 175–226.

    Google Scholar 

  9. Lorimer, J. W. and Buchholz, M. (1992) Homomorphisms of affine and projective planes,Math. Magazine 55, 195–204.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Ottawa, K1N 6N5, Ottawa, Canada

    Ken Grant

  2. Department of Mathematics, Dalhousie University, Halifax, Canada

    R. J. Nowakowski

  3. Department of Computer Science, University of Ottawa, K1N 6N5, Ottawa, Canada

    Ivan Rival

Authors
  1. Ken Grant
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  2. R. J. Nowakowski
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  3. Ivan Rival
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Additional information

Communicated by P. Hell

Partially supported by a grant from the NSERC.

Partially supported by a grant from the NSERC.

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Grant, K., Nowakowski, R.J. & Rival, I. The endomorphism spectrum of an ordered set. Order 12, 45–55 (1995). https://doi.org/10.1007/BF01108589

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  • DOI: https://doi.org/10.1007/BF01108589

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