Difference between revisions of "Pointwise order"
From Encyclopedia of Mathematics
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| − | The [[Order (on a set)|order]] on the | + | 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 |
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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==== | ||
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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> | + | <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
Pointwise order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointwise_order&oldid=36146