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Cardinal function ：ウィキペディア英語版

Cardinal function
In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers.
== Cardinal functions in set theory ==

* The most frequently used cardinal function is a function which assigns to a set "A" its cardinality, denoted by | ''A'' |.
* Aleph numbers and beth numbers can both be seen as cardinal functions defined on ordinal numbers.
* Cardinal arithmetic operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers.
* Cardinal characteristics of a (proper) ideal ''I'' of subsets of ''X'' are:
:

(I)=\min\\subseteq I \wedge \bigcup\notin I\big\}.

::The "additivity" of ''I'' is the smallest number of sets from ''I'' whose union is not in ''I'' any more. As any ideal is closed under finite unions, this number is always at least

\aleph_0

; if ''I'' is a σ-ideal, then

\operatorname(I) \ge \aleph_1.

\operatorname(I)=\min\\subseteq I \wedge\bigcup=X\big\}.

:: The "covering number" of ''I'' is the smallest number of sets from ''I'' whose union is all of ''X''. As ''X'' itself is not in ''I'', we must have add(''I'') ≤ cov(''I'').
:

\operatorname(I)=\min\,

:: The "uniformity number" of ''I'' (sometimes also written

(I)

) is the size of the smallest set not in ''I''. Assuming ''I'' contains all singletons, add(''I'') ≤ non(''I'').
:

(I)=\min\\subseteq I \wedge (\forall A\in I)(\exists B\in )(A\subseteq B)\big\}.

:: The "cofinality" of ''I'' is the cofinality of the partial order (''I'', ⊆). It is easy to see that we must have non(''I'') ≤ cof(''I'') and cov(''I'') ≤ cof(''I'').
:In the case that

I

is an ideal closely related to the structure of the reals, such as the ideal of Lebesgue null sets or the ideal of meagre sets, these cardinal invariants are referred to as cardinal characteristics of the continuum.
* For a preordered set

(,\sqsubseteq)

the bounding number

()

and dominating number

()

is defined as
::

()=\min\big\)(\exists y\in Y)(y\not\sqsubseteq x)\big\},

()=\min\big\)(\exists y\in Y)(x\sqsubseteq y)\big\}.

* In PCF theory the cardinal function

pp_\kappa(\lambda)

is used.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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