Antichain

Inmathematics,in the area oforder theory,anantichainis a subset of apartially ordered setsuch 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, theheightof the partially ordered set (the length of its longest chain) equals byMirsky's theoremthe 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 meetoperations, 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 families and their lattice is afree distributive lattice,with aDedekind numberof elements. More generally, counting the number of antichains of a finite partially ordered set is#P-complete.

Definitions

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

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

Height and width

Amaximal antichainis an antichain that is not aproper subsetof any other antichain. Amaximum antichainis an antichain that has cardinality at least as large as every other antichain. Thewidthof 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 $k$ chains then the width of the order must be at most $k$ (if the antichain has more than $k$ elements, by thepigeonhole principle,there would be 2 of its elements belonging to the same chain, a contradiction).Dilworth's theoremstates 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 theheightof a partial order to be the maximum cardinality of a chain.Mirsky's theoremstates that in any partial order of finite height, the height equals the smallest number of antichains into which the order may be partitioned.^[2]

Sperner families

An antichain in the inclusion ordering of subsets of an $n$ -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 (sequenceA000372in 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.

Join and meet operations

Any antichain $A$ corresponds to alower set $L_{A}=\{x:\exists y\in A{\mbox{ such that }}x\leq y\}.$ 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 ajoinoperation on antichains: $A\vee B=\{x\in A\cup B:\nexists y\in A\cup B{\mbox{ such that }}x<y\}.$ Similarly, we can define ameetoperation on antichains, corresponding to the intersection of lower sets: $A\wedge B=\{x\in L_{A}\cap L_{B}:\nexists y\in L_{A}\cap L_{B}{\mbox{ such that }}x<y\}.$ The join and meet operations on all finite antichains of finite subsets of a set $X$ define adistributive lattice,the free distributive lattice generated by $X.$ Birkhoff's representation theoremfor 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 setsof the partial order.^[4]

Computational complexity

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]

References

^Dilworth, Robert P.(1950), "A decomposition theorem for partially ordered sets",Annals of Mathematics,51(1): 161–166,doi:10.2307/1969503,JSTOR 1969503
^Mirsky, Leon(1971), "A dual of Dilworth's decomposition theorem",American Mathematical Monthly,78(8): 876–877,doi:10.2307/2316481,JSTOR 2316481
^Kahn, Jeff (2002), "Entropy, independent sets and antichains: a new approach to Dedekind's problem",Proceedings of the American Mathematical Society,130(2): 371–378,doi:10.1090/S0002-9939-01-06058-0,MR 1862115
^Birkhoff, Garrett(1937), "Rings of sets",Duke Mathematical Journal,3(3): 443–454,doi:10.1215/S0012-7094-37-00334-X
^Felsner, Stefan; Raghavan, Vijay; Spinrad, Jeremy (2003),"Recognition algorithms for orders of small width and graphs of small Dilworth number",Order,20(4): 351–364 (2004),doi:10.1023/B:ORDE.0000034609.99940.fb,MR 2079151,S2CID 1363140
^Provan, J. Scott; Ball, Michael O. (1983), "The complexity of counting cuts and of computing the probability that a graph is connected",SIAM Journal on Computing,12(4): 777–788,doi:10.1137/0212053,MR 0721012

External links

[dilworth-1] Dilworth, Robert P.(1950), "A decomposition theorem for partially ordered sets",Annals of Mathematics,51(1): 161–166,doi:10.2307/1969503,JSTOR 1969503

[mirsky-2] Mirsky, Leon(1971), "A dual of Dilworth's decomposition theorem",American Mathematical Monthly,78(8): 876–877,doi:10.2307/2316481,JSTOR 2316481

[kahn-3] Kahn, Jeff (2002), "Entropy, independent sets and antichains: a new approach to Dedekind's problem",Proceedings of the American Mathematical Society,130(2): 371–378,doi:10.1090/S0002-9939-01-06058-0,MR 1862115

[birkhoff-4] Birkhoff, Garrett(1937), "Rings of sets",Duke Mathematical Journal,3(3): 443–454,doi:10.1215/S0012-7094-37-00334-X

[felragspin-5] Felsner, Stefan; Raghavan, Vijay; Spinrad, Jeremy (2003),"Recognition algorithms for orders of small width and graphs of small Dilworth number",Order,20(4): 351–364 (2004),doi:10.1023/B:ORDE.0000034609.99940.fb,MR 2079151,S2CID 1363140

[provball-6] Provan, J. Scott; Ball, Michael O. (1983), "The complexity of counting cuts and of computing the probability that a graph is connected",SIAM Journal on Computing,12(4): 777–788,doi:10.1137/0212053,MR 0721012

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