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  • 1 Department of Mathematics, , Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, , United States of America
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Let k ≥ 1. A Sperner k-family is a maximum-sized subset of a finite poset that contains no chain with k + 1 elements. In 1976 Greene and Kleitman defined a lattice-ordering on the set Sk(P) of Sperner k-families of a fifinite poset P and posed the problem: “Characterize and interpret the join- and meet-irreducible elements of Sk(P),” adding, “This has apparently not been done even for the case k = 1.”

In this article, the case k = 1 is done.

  • [1]

    Davey, B. A. and Priestley, H. A. Introduction to Lattices and Order. Cambridge University Press, Cambridge, 2002. Second edition.

  • [2]

    Dilworth, R. P. A decomposition theorem for partially ordered sets. Annals of Mathematics 51 (1950), 161166.

  • [3]

    Dilworth, R. P. Some combinatorial problems on partially ordered sets. In Proceedings of Symposia in Applied Mathematics (Providence, Rhode Island, 1960), vol. 10., American Mathematical Society, pages. 8590.

    • Search Google Scholar
    • Export Citation
  • [4]

    Freese, R. An application of Dilworth’s lattice of maximal antichains. Discrete Mathematics 7 (1974), 107109.

  • [5]

    Freese, R., Jezcaronek, J. and Nation, J. B. Free Lattices. American Mathematical Society, Providence, Rhode Island, 1995.

  • [6]

    Greene, C. and Kleitman, D. J. The structure of Sperner k-families. Journal of Combinatorial Theory (A) 20 (1976), 4168.

  • [7]

    Hall, P. On representatives of subsets. Journal of the London Mathematical Society 10 (1935), 2630.

  • [8]

    Koh, K. M. On the lattice of maximum-sized antichains of a finite poset. Algebra Universalis, 17 (1983), 7386.

  • [9]

    Wikipedia. Strongly connected component. (December 26, 2015). Retrieved from https://en.wikipedia.org/wiki/Strongly_connected_component.

    • Search Google Scholar
    • Export Citation
  • [10]

    Wild, M. Letters to author, December 3 and 10, 2015.

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Mathematica Pannonica
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