J u m p t o c o n t e n t
M a i n m e n u
M a i n m e n u
N a v i g a t i o n
● M a i n p a g e
● C o n t e n t s
● C u r r e n t e v e n t s
● R a n d o m a r t i c l e
● A b o u t W i k i p e d i a
● C o n t a c t u s
● D o n a t e
C o n t r i b u t e
● H e l p
● L e a r n t o e d i t
● C o m m u n i t y p o r t a l
● R e c e n t c h a n g e s
● U p l o a d f i l e
S e a r c h
Search
A p p e a r a n c e
● C r e a t e a c c o u n t
● L o g i n
P e r s o n a l t o o l s
● C r e a t e a c c o u n t
● L o g i n
P a g e s f o r l o g g e d o u t e d i t o r s l e a r n m o r e
● C o n t r i b u t i o n s
● T a l k
( T o p )
1
S e e a l s o
2
R e f e r e n c e s
T o g g l e t h e t a b l e o f c o n t e n t s
P r o d u c t o r d e r
2 l a n g u a g e s
● 日 本 語
● த ம ி ழ ்
E d i t l i n k s
● A r t i c l e
● T a l k
E n g l i s h
● R e a d
● E d i t
● V i e w h i s t o r y
T o o l s
T o o l s
A c t i o n s
● R e a d
● E d i t
● V i e w h i s t o r y
G e n e r a l
● W h a t l i n k s h e r e
● R e l a t e d c h a n g e s
● U p l o a d f i l e
● S p e c i a l p a g e s
● P e r m a n e n t l i n k
● P a g e i n f o r m a t i o n
● C i t e t h i s p a g e
● G e t s h o r t e n e d U R L
● D o w n l o a d Q R c o d e
● W i k i d a t a i t e m
P r i n t / e x p o r t
● D o w n l o a d a s P D F
● P r i n t a b l e v e r s i o n
A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Hasse diagram of the product order on
N
{\displaystyle \mathbb {N} }
×
N
{\displaystyle \mathbb {N} }
In mathematics , given a partial order
⪯
{\displaystyle \preceq }
and
⊑
{\displaystyle \sqsubseteq }
on a set
A
{\displaystyle A}
and
B
{\displaystyle B}
, respectively, the product order [1] [2] [3] [4] (also called the coordinatewise order [5] [3] [6] or componentwise order [2] [7] ) is a partial ordering
≤
{\displaystyle \leq }
on the Cartesian product
A
×
B
.
{\displaystyle A\times B.}
Given two pairs
(
a
1
,
b
1
)
{\displaystyle \left(a_{1},b_{1}\right)}
and
(
a
2
,
b
2
)
{\displaystyle \left(a_{2},b_{2}\right)}
in
A
×
B
,
{\displaystyle A\times B,}
declare that
(
a
1
,
b
1
)
≤
(
a
2
,
b
2
)
{\displaystyle \left(a_{1},b_{1}\right)\leq \left(a_{2},b_{2}\right)}
if
a
1
⪯
a
2
{\displaystyle a_{1}\preceq a_{2}}
and
b
1
⊑
b
2
.
{\displaystyle b_{1}\sqsubseteq b_{2}.}
Another possible ordering on
A
×
B
{\displaystyle A\times B}
is the lexicographical order . It is a total ordering if both
A
{\displaystyle A}
and
B
{\displaystyle B}
are totally ordered. However the product order of two total orders is not in general total; for example, the pairs
(
0
,
1
)
{\displaystyle (0,1)}
and
(
1
,
0
)
{\displaystyle (1,0)}
are incomparable in the product order of the ordering
0
<
1
{\displaystyle 0<1}
with itself. The lexicographic combination of two total orders is a linear extension of their product order, and thus the product order is a subrelation of the lexicographic order.[3]
The Cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions .[7]
The product order generalizes to arbitrary (possibly infinitary) Cartesian products.
Suppose
A
≠
∅
{\displaystyle A\neq \varnothing }
is a set and for every
a
∈
A
,
{\displaystyle a\in A,}
(
I
a
,
≤
)
{\displaystyle \left(I_{a},\leq \right)}
is a preordered set.
Then the product preorder on
∏
a
∈
A
I
a
{\displaystyle \prod _{a\in A}I_{a}}
is defined by declaring for any
i
∙
=
(
i
a
)
a
∈
A
{\displaystyle i_{\bullet }=\left(i_{a}\right)_{a\in A}}
and
j
∙
=
(
j
a
)
a
∈
A
{\displaystyle j_{\bullet }=\left(j_{a}\right)_{a\in A}}
in
∏
a
∈
A
I
a
,
{\displaystyle \prod _{a\in A}I_{a},}
that
i
∙
≤
j
∙
{\displaystyle i_{\bullet }\leq j_{\bullet }}
if and only if
i
a
≤
j
a
{\displaystyle i_{a}\leq j_{a}}
for every
a
∈
A
.
{\displaystyle a\in A.}
If every
(
I
a
,
≤
)
{\displaystyle \left(I_{a},\leq \right)}
is a partial order then so is the product preorder.
Furthermore, given a set
A
,
{\displaystyle A,}
the product order over the Cartesian product
∏
a
∈
A
{
0
,
1
}
{\displaystyle \prod _{a\in A}\{0,1\}}
can be identified with the inclusion ordering of subsets of
A
.
{\displaystyle A.}
[4]
The notion applies equally well to preorders . The product order is also the categorical product in a number of richer categories, including lattices and Boolean algebras .[7]
See also [ edit ]
References [ edit ]
^ a b Sudhir R. Ghorpade; Balmohan V. Limaye (2010). A Course in Multivariable Calculus and Analysis . Springer. p. 5. ISBN 978-1-4419-1621-1 .
^ a b c Egbert Harzheim (2006). Ordered Sets . Springer. pp. 86–88. ISBN 978-0-387-24222-4 .
^ a b Victor W. Marek (2009). Introduction to Mathematics of Satisfiability . CRC Press. p. 17. ISBN 978-1-4398-0174-1 .
^ Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 18
^ Alexander Shen; Nikolai Konstantinovich Vereshchagin (2002). Basic Set Theory . American Mathematical Soc. p. 43. ISBN 978-0-8218-2731-4 .
^ a b c Paul Taylor (1999). Practical Foundations of Mathematics . Cambridge University Press. pp. 144–145 and 216. ISBN 978-0-521-63107-5 .
t
e
Key concepts
Results
Properties & Types (list )
Constructions
Topology & Orders
Related
t
e
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Product_order&oldid=1221121226 "
C a t e g o r i e s :
● O r d e r t h e o r y
● M a t h e m a t i c s s t u b s
H i d d e n c a t e g o r y :
● A l l s t u b a r t i c l e s
● T h i s p a g e w a s l a s t e d i t e d o n 2 8 A p r i l 2 0 2 4 , a t 0 0 : 4 1 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
● D e v e l o p e r s
● S t a t i s t i c s
● C o o k i e s t a t e m e n t
● M o b i l e v i e w