Sperner property of a partially ordered set

In order-theoretic mathematics, a graded partially ordered set is said to have the Sperner property (and hence is called a Sperner poset), if no antichain within it is larger than the largest rank level (one of the sets of elements of the same rank) in the poset.{{citation

| last = Stanley | first = Richard

|authorlink=Richard P. Stanley

| doi = 10.1007/BF00396271

| mr = 0745587

| issue = 1

| journal = Order

| pages = 29–34

| title = Quotients of Peck posets

| volume = 1

| year = 1984| s2cid = 14857863

}}. Since every rank level is itself an antichain, the Sperner property is equivalently the property that some rank level is a maximum antichain.[https://books.google.com/books?id=yJIMx9nXB6kC&dq=%22graded+poset%22+rank&pg=PA723 Handbook of discrete and combinatorial mathematics, by Kenneth H. Rosen, John G. Michaels] The Sperner property and Sperner posets are named after Emanuel Sperner, who proved Sperner's theorem stating that the family of all subsets of a finite set (partially ordered by set inclusion) has this property. The lattice of partitions of a finite set typically lacks the Sperner property.{{citation

| last = Graham | first = R. L. | authorlink = Ronald Graham

| date = June 1978

| doi = 10.1007/BF03023067

| issue = 2

| journal = The Mathematical Intelligencer

| mr = 0505555

| pages = 84–86

| title = Maximum antichains in the partition lattice

| url = https://www.math.ucsd.edu/~ronspubs/78_14_antichains.pdf

| volume = 1| s2cid = 120190991 }}