In
mathematics, given two
preordered sets
and
the product order
(also called the coordinatewise order
[Davey & Priestley, '' Introduction to Lattices and Order'' (Second Edition), 2002, p. 18] or componentwise order
) is a partial ordering on the
Cartesian product Given two pairs
and
in
declare that
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
and
Another possible ordering on
is the
lexicographical order, which is a
total ordering
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
. However the product order of two
totally ordered
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
sets is not in general total; for example, the pairs
and
are incomparable in the product order of the ordering
with itself. The lexicographic order of totally ordered sets is a
linear extension of their product order, and thus the product order is a
subrelation of the lexicographic order.
The Cartesian product with the product order is the
categorical product in the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of partially ordered sets with
monotone function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
s.
The product order generalizes to arbitrary (possibly infinitary) Cartesian products.
Suppose
is a set and for every
is a preordered set.
Then the on
is defined by declaring for any
and
in
that
:
if and only if
for every
If every
is a partial order then so is the product preorder.
Furthermore, given a set
the product order over the Cartesian product
can be identified with the inclusion ordering of subsets of
The notion applies equally well to
preorders. The product order is also the categorical product in a number of richer categories, including
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
s and
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
s.
References
See also
*
Direct product of binary relations
*
Examples of partial orders
*
Star product, a different way of combining partial orders
*
Orders on the Cartesian product of totally ordered sets
*
Ordinal sum
In mathematics, especially order theory, a partially ordered set (also poset) 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 r ...
of partial orders
*
{{math-stub
Order theory