Mathematica Pannonica
Volume/Issue:
Volume 27_NS1: Issue 2

The “k = 1” Case of a Problem of Greene and Kleitman from 1976: Join-Irreducible Elements in the Lattice of Sperner 1-Families

Author:
Jonathan David Farley Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, United States of America

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Pages:
95–104
Online Publication Date:
27 Sep 2021
Publication Date:
18 Nov 2021
Article Category:
Research Article
DOI:
https://doi.org/10.1556/314.2021.00010
Keywords:
Sperner k-family; antichain; join-irreducible element; Dilworth’s Theorem; Hall’s Marriage Theorem; (partially) ordered set
Open access

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.

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Mathematica Pannonica
Print ISSN:
0865-2090
Online ISSN:
2786-0752
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Mathematica Pannonica
Language English
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ISSN 2786-0752 (Online)
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