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Difference between revisions of "Pointwise order"

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(cite Davey & Priestley (2002))
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The [[Order (on a set)|order]] on the section of functions with values in a [[partially ordered set]].  If $f$ and $g$ are functions from $X$ to $Y$, where $(Y,<)$ is ordered, then there is an [[order relation]] on $Y^X$ defined by
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The [[Order (on a set)|order]] on the set of functions with values in a [[partially ordered set]].  If $f$ and $g$ are functions from $X$ to $Y$, where $(Y,<)$ is ordered, then there is an [[order relation]] on $Y^X$ defined by
 
$$
 
$$
 
f \le g \Leftrightarrow \forall x \in X\,,\ f(x) \le g(x) \ .
 
f \le g \Leftrightarrow \forall x \in X\,,\ f(x) \le g(x) \ .
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====References====
 
====References====
 
<table>
 
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  B. A. Davey, H. A. Priestley, ''Introduction to lattices and order'', 2nd ed. Cambridge University Press  (2002) ISBN 978-0-521-78451-1 {{ZBL|1002.06001}}</TD></TR>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  B. A. Davey, H. A. Priestley, ''Introduction to lattices and order'', 2nd ed. Cambridge University Press  (2002) {{ISBN|978-0-521-78451-1}} {{ZBL|1002.06001}}</TD></TR>
 
</table>
 
</table>

Latest revision as of 07:33, 24 November 2023

The order on the set of functions with values in a partially ordered set. If $f$ and $g$ are functions from $X$ to $Y$, where $(Y,<)$ is ordered, then there is an order relation on $Y^X$ defined by $$ f \le g \Leftrightarrow \forall x \in X\,,\ f(x) \le g(x) \ . $$

See also: Pointwise operation.

References

[1] B. A. Davey, H. A. Priestley, Introduction to lattices and order, 2nd ed. Cambridge University Press (2002) ISBN 978-0-521-78451-1 Zbl 1002.06001
How to Cite This Entry:
Pointwise order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointwise_order&oldid=36146