product of posets
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 .
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,
iff there is some such that for all and .
We show that this is indeed a partial order on :
The three things we need to verify are
(Reflexivity). Clearly, , since for any .
(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|
|Date of creation||2013-03-22 16:33:25|
|Last modified on||2013-03-22 16:33:25|
|Last modified by||CWoo (3771)|
|Defines||product of lattices|