product of posets

Cartesian Ordering

Let (P1,1) and (P2,2) be posets. Let P=P1×P2, the Cartesian productMathworldPlanetmath of the underlying sets. Next, define a binary relationMathworldPlanetmath on P, given by

(a,b)(c,d)   iff   a1c and b2d.

Then is a partial orderMathworldPlanetmath on P. (P,) is called the product of posets (P1,1) and (P2,2). The ordering is called the Cartesian ordering. As it is customary, we write P to mean (P,).

If P1 and P2 are antichainsMathworldPlanetmath, their productMathworldPlanetmathPlanetmath is also an antichain. If they are both join semilattices, then their product P is a join semilattice as well. The join of (a,b) and (c,d) are given by


Conversely, if the product P of two posets P1 and P2 is a join semilattice, then P1 and P2 are both join semilattices. If (a,b)(c,d)=(e,f), then e is the upper bound of a and c. If ge is an upper bound of a and c, then (g,f) is an upper bound of (a,b) and (c,d), whence (g,f)=(e,f), or g=e. So g=a1c. Similarly, f=b2d. Dually, P=P1×P2 is a meet semilattice (and consequently, a latticeMathworldPlanetmath) iff both P1 and P2 are. Equivalently, the product of (semi)lattices can be defined purely algebraically (using and only).

Another simple fact about the product of posets is the following: the product is never a chain unless one of the posets is trivial (a singleton). To see this, let P=P1×P2 and (a,b),(c,d)P. Then (a,b) and (c,d) are comparablePlanetmathPlanetmath, say (a,b)(c,d), which implies a1c and b2d. Also, (c,b) and (a,d) are comparable. But since b2d, we must have (c,b)(a,d), which means c1a, showing a=c, or P1={a}.

Remark. The product of two posets can be readily extended to any finite product, countably infiniteMathworldPlanetmath product, or even arbitrary product of posets. The definition is similar to the one given above and will not be repeated here.

An example of a product of posets is the lattice in n (, which is defined as the free abelian groupMathworldPlanetmath over in n generatorsPlanetmathPlanetmathPlanetmath. But from a poset perspective, it can be viewed as a product of n chains, each order isomorphic to . As we have just seen earlier, this product is a lattice, and hence the name “lattice” in n.

Lexicographic Ordering

Again, let P1 and P2 be posets. Form the Cartesian product of P1 and P2 and call it P. There is another way to partial order P, called the lexicographic orderMathworldPlanetmath. Specifically,

(a,b)(c,d) iff {ac, or a=c and bd.

More generally, if {PiiI} is a collectionMathworldPlanetmath of posets indexed by a set I that is linearly orderedPlanetmathPlanetmath, then the Cartesian product P:=Pi also has the lexicographic order:

(ai)(bi) iff there is some kI such that aj=bj for all j<k and akbk.

We show that this is indeed a partial order on P:


The three things we need to verify are

  • (ReflexivityMathworldPlanetmath). Clearly, (ai)(ai), since aiai for any iI.

  • (Transitivity). If (ai)(bi) and (bi)(ci), then for some k,I we have that

    1. (a)

      aj=bj for all j<k and akbk, and

    2. (b)

      bj=cj for all j< and bc.

    Since I is a total order, k and are comparable, say k, so that aj=bj=cj for all k< and akbkck. Since Pk is partially ordered, akck as well. Therefore (ai)(ci).

  • (Antisymmetry). Finally, suppose (ai)(bi) and (bi)(ai). If (ai)(bi), then (ai)(bi) implies that we can find kI such that aj=bj for all j<k and ak<bk. By the same token, (bi)(ai) implies the existence of I with bj=aj for all j< and b<a. Since I is linearly ordered, we can again assume that k. But then this means that either k<, in which case bk=ak, a contradictionMathworldPlanetmathPlanetmath, or k=, in which case we have that ak<bk=b<a=ak, another contradiction. Therefore (ai)=(bi).

This completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof. ∎

Title product of posets
Canonical name ProductOfPosets
Date of creation 2013-03-22 16:33:25
Last modified on 2013-03-22 16:33:25
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 14
Author CWoo (3771)
Entry type Definition
Classification msc 06A06
Classification msc 06A99
Related topic LatticeInMathbbRn
Defines product of lattices
Defines Cartesian ordering
Defines lexicographic ordering