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Multiplicative posets

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Abstract

A functionf from the posetP to the posetQ is a strict morphism if for allx, y ∈ P withx<y we havef(x)<f(y). If there is such a strict morphism fromP toQ we writeP → Q, otherwise we writeP \(\not \to \) Q. We say a posetM is multiplicative if for any posetsP, Q withP \(\not \to \) M andQ \(\not \to \) M we haveP ×Q \(\not \to \) M. (Here (p 1,q 1)<(p 2,q 2) if and only ifp 1<p 2 andq 1<q 2.) This paper proves that well-founded trees with height ≤ω are multiplicative posets.

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References

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    M.El-Zahar and N. W.Sauer (1985) The chromatic number of the products of two 4-chromatic graphs is 4,Combinatorica 5(2), 121–126.

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    R.Haggkvist, P.Hell, D. J.Miller, and V. N.Lara (1988) On multiplicative graphs and the product conjecture,Combinatorica 8(1), 63–74.

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    S. T. Hedetniemi (1966) Homomorphisms of graphs and automata, University of Michigan Technical Report 03105-44-T.

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    P. Hell, H. Zhou, and X. Zhu (1991) Multiplicative oriented cycles. Manuscript.

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Additional information

This research was supported in part by NSERC Grant #69-1325.

Communicated by E. C. Milner

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Sauer, N.W., Zhu, X. Multiplicative posets. Order 8, 349–358 (1991). https://doi.org/10.1007/BF00571185

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Mathematics Subject Classification (1991)

  • 06A06

Key words

  • Posets
  • strict morphisms
  • multiplicativity
  • Hedetniemi's conjecture