product of posets
Cartesian Ordering
Let and be posets. Let , the Cartesian product of the underlying sets. Next, define a binary relation on , given by
Then is a partial order on . is called the product of posets and . The ordering is called the Cartesian ordering. As it is customary, we write to mean .
If and are antichains, their product is also an antichain. If they are both join semilattices, then their product is a join semilattice as well. The join of and are given by
Conversely, if the product of two posets and is a join semilattice, then and are both join semilattices. If , then is the upper bound of and . If is an upper bound of and , then is an upper bound of and , whence , or . So . Similarly, . Dually, is a meet semilattice (and consequently, a lattice) iff both and 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 and . Then and are comparable, say , which implies and . Also, and are comparable. But since , we must have , which means , showing , or .
Remark. The product of two posets can be readily extended to any finite product, countably infinite 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 (http://planetmath.org/LatticeInMathbbRn), which is defined as the free abelian group over in generators. But from a poset perspective, it can be viewed as a product of chains, each order isomorphic to . As we have just seen earlier, this product is a lattice, and hence the name “lattice” in .
Lexicographic Ordering
Again, let and be posets. Form the Cartesian product of and and call it . There is another way to partial order , called the lexicographic order. Specifically,
More generally, if is a collection of posets indexed by a set that is linearly ordered, then the Cartesian product also has the lexicographic order:
iff there is some such that for all and .
We show that this is indeed a partial order on :
Proof.
The three things we need to verify are
-
•
(Reflexivity). Clearly, , since for any .
-
•
(Transitivity). If and , then for some we have that
-
(a)
for all and , and
-
(b)
for all and .
Since is a total order, and are comparable, say , so that for all and . Since is partially ordered, as well. Therefore .
-
(a)
-
•
(Antisymmetry). Finally, suppose and . If , then implies that we can find such that for all and . By the same token, implies the existence of with for all and . Since is linearly ordered, we can again assume that . But then this means that either , in which case , a contradiction, or , in which case we have that , another contradiction. Therefore .
This completes 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 |