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On Jump-Critical Posets with Jump-Number Equal to Width

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Abstract

We investigate the structure of m-jump-critical posets P with w(P) = m. We prove that the size of such posets satisfies |P| ≤ 3m 2. For the special when the maximum antichain occurs as the maximal (or minimal) elements, we have the sharp upper bound |P| ≤ 3m - k; where k = min {|{Max}(P)|, |{Min}(P)|}. We give examples of posets which illustrate the explored structure of these m-jump-critical posets P with width m.

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References

  1. El-Zahar, M. H. and Schmerl, J. H. (1984) On the size of jump-critical ordered sets, Order 1, 3-5.

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  2. El-Zahar, M. H. and Rival, I. (1985) Examples of jump-critical ordered sets, SIAM J. Algebraic Discrete Methods 6(4), 713-720.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Ain Shams University, Abbaseia, Cairo, Egypt

    Mohamed H. El-Zahar

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  1. Mohamed H. El-Zahar
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El-Zahar, M.H. On Jump-Critical Posets with Jump-Number Equal to Width. Order 17, 93–101 (2000). https://doi.org/10.1023/A:1006401802560

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  • DOI: https://doi.org/10.1023/A:1006401802560

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