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In [[mathematics]], specifically [[set theory]], a '''dimensional operator''' on a set ''E'' is a function from the [[subsets]] of ''E'' to the subsets of ''E''.


==Definition==


If the [[power set]] of ''E'' is denoted ''P''(''E'') then a dimensional operator on ''E'' is a map
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:<math>d:P(E)\rightarrow P(E)\,</math>
that satisfies the following properties for ''S'',''T'' &isin; ''P''(''E''):
# ''S'' &sube; ''d''(''S'');
# ''d''(''S'') = ''d''(''d''(''S'')) (''d'' is [[idempotent]]);
# if ''S'' &sube; ''T'' then ''d''(''S'') &sube; ''d''(''T'');
# if Ω is the set of finite subsets of ''S'' then ''d''(''S'') = &cup;<sub>''A''&isin;Ω</sub>''d''(''A'');
# if ''x'' &isin; ''E'' and ''y'' &isin; ''d''(''S'' &cup; {''x''}) \ ''d''(''S''), then ''x'' &isin; ''d''(''S'' &cup; {''y''}).
 
The final property is known as the exchange axiom.<ref>Julio R. Bastida, ''Field Extensions and Galois Theory'', Addison-Wesley Publishing Company, 1984, pp.&nbsp;212&ndash;213.</ref>
 
==Examples==
 
# For any set ''E'' the identity map on ''P''(''E'') is a dimensional operator.
# The map which takes any subset ''S'' of ''E'' to ''E'' itself is a dimensional operator on ''E''.
 
==References==
<references/>
 
[[Category:Set theory]]

Latest revision as of 23:52, 11 November 2014


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