Product order
Encyclopedia
In mathematics
, given two ordered set
s A and B, one can induce a partial ordering on the Cartesian product
A × B. Given
two pairs (a1,b1) and (a2,b2) in A × B, one sets
≤ (a2,b2)
if and only if
a1 ≤ a2 and
b1 ≤ b2.
This ordering is called the product order. Another possible ordering on A × B is the lexicographical order
.
The Cartesian product with product order is the categorical product in the category
of partially ordered sets with monotone functions.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, given two ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
s A and B, one can induce a partial ordering on the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
A × B. Given
two pairs (a1,b1) and (a2,b2) in A × B, one sets
≤ (a2,b2)
if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
a1 ≤ a2 and
b1 ≤ b2.
This ordering is called the product order. Another possible ordering on A × B is the lexicographical order
Lexicographical order
In mathematics, the lexicographic or lexicographical order, , is a generalization of the way the alphabetical order of words is based on the alphabetical order of letters.-Definition:Given two partially ordered sets A and B, the lexicographical order on...
.
The Cartesian product with product order is the categorical product in the category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of partially ordered sets with monotone functions.
See also
- direct product of binary relations
- examples of partial orders
- orders on the Cartesian product of totally ordered sets