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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  S k  ( P ) of Sperner  k -families of a fifinite poset  P  and posed the problem: “Characterize and interpret the join- and meet-irreducible elements of  S k  ( P ),” adding, “This has apparently not been done even for the case  k  = 1.”  In this article, the case  k  = 1 is done.

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