Inmathematics, in the area oforder theory, anantichain is a subset of apartially ordered set such that any two distinct elements in the subset areincomparable.

The size of the largest antichain in a partially ordered set is known as itswidth. ByDilworth's theorem, this also equals the minimum number of chains (totally ordered subsets) into which the set can be partitioned. Dually, theheight of the partially ordered set (the length of its longest chain) equals byMirsky's theorem the minimum number of antichains into which the set can be partitioned.

The family of all antichains in a finite partially ordered set can be givenjoin and meet operations, making them into adistributive lattice. For the partially ordered system of all subsets of a finite set, ordered by set inclusion, the antichains are calledSperner familiesand their lattice is afree distributive lattice, with aDedekind number of elements. More generally, counting the number of antichains of a finite partially ordered set is#P-complete.

Let${\displaystyle S}$ be a partially ordered set. Two elements${\displaystyle a}$ and${\displaystyle b}$ of a partially ordered set are calledcomparable if${\displaystyle a\le b\text{ or }b\le a.}$ If two elements are not comparable, they are called incomparable; that is,${\displaystyle x}$ and${\displaystyle y}$ are incomparable if neither${\displaystyle x\le y\text{ nor }y\le x.}$

A chain in${\displaystyle S}$ is asubset${\displaystyle C\subseteq S}$ in which each pair of elements is comparable; that is,${\displaystyle C}$ istotally ordered. An antichain in${\displaystyle S}$ is asubset${\displaystyle A}$ of${\displaystyle S}$ in which each pair of different elements is incomparable; that is, there is no order relation between any two different elements in${\displaystyle A.}$ (However, some authors use the term "antichain" to meanstrong antichain, a subset such that there is no element of theposet smaller than two distinct elements of the antichain.)

Amaximal antichain is an antichain that is not aproper subset of any other antichain. Amaximum antichain is an antichain that has cardinality at least as large as every other antichain. Thewidth of a partially ordered set is the cardinality of a maximum antichain. Any antichain can intersect any chain in at most one element, so, if we can partition the elements of an order into${\displaystyle k}$ chains then the width of the order must be at most${\displaystyle k}$ (if the antichain has more than${\displaystyle k}$ elements, by thepigeonhole principle, there would be 2 of its elements belonging to the same chain, a contradiction).Dilworth's theorem states that this bound can always be reached: there always exists an antichain, and a partition of the elements into chains, such that the number of chains equals the number of elements in the antichain, which must therefore also equal the width.^[1] Similarly, one can define theheight of a partial order to be the maximum cardinality of a chain.Mirsky's theorem states that in any partial order of finite height, the height equals the smallest number of antichains into which the order may be partitioned.^[2]

An antichain in the inclusion ordering of subsets of an${\displaystyle n}$-element set is known as aSperner family. The number of different Sperner families is counted by theDedekind numbers,^[3] the first few of which numbers are

2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 (sequenceA000372 in theOEIS).

Even the empty set has two antichains in its power set: one containing a single set (the empty set itself) and one containing no sets.

Any antichain${\displaystyle A}$ corresponds to alower set$${\displaystyle L_A=\{x:\mathrm{\exists }y\in A\text{ such that }x\le y\}.}$$In a finite partial order (or more generally a partial order satisfying theascending chain condition) all lower sets have this form. The union of any two lower sets is another lower set, and the union operation corresponds in this way to ajoin operationon antichains:Similarly, we can define ameet operation on antichains, corresponding to the intersection of lower sets:The join and meet operations on all finite antichains of finite subsets of a set${\displaystyle X}$ define adistributive lattice, the free distributive lattice generated by${\displaystyle X.}$Birkhoff's representation theorem for distributive lattices states that every finite distributive lattice can be represented via join and meet operations on antichains of a finite partial order, or equivalently as union and intersection operations on thelower sets of the partial order.^[4]

A maximum antichain (and its size, the width of a given partially ordered set) can be found inpolynomial time.^[5]Counting the number of antichains in a given partially ordered set is#P-complete.^[6]

• Weisstein, Eric W."Antichain".MathWorld.
• "Antichain".PlanetMath.

• Order theory

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