Equality in a result of Kleitman

Daniel McQuillan, R.Bruce Richter
1994 Journal of combinatorial theory. Series A  
An upset is a set q/ of subset of a finite set. S such that if U~ V and Ueq/, then V~ql. A downset 9 is defined analogously. In 1966, Kleitman (J. Combin. Theory 1 (1966), 153-155) proved that if q/and 9 are arbitrary up-and downsets, respectively, then [og[ [9[/> 2 Isl [og c~ 9[. In this note, we show that a necessary and sufficient condition for equality to hold is: for every minimal element U of og and every maximal element D of 9, U_ D. This result is extended to some related inequalities.
doi:10.1016/0097-3165(94)90029-9 fatcat:dwd2nnabj5d2pbq67mgpxusjgm