Cardinal function

In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers.

In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers. {\rm add}(I) = \min\ e(X) \leq s(X) \chi(X) \leq w(X) \operatorname{nw}(X) \leq w(X) \text{ and } o(X) \leq 2^{\operatorname{nw}(X)} ==Cardinal functions in Boolean algebras== Cardinal functions are often used in the study of Boolean algebras. We can mention, for example, the following functions: Cellularity c(\mathbb{B}) of a Boolean algebra \mathbb{B} is the supremum of the cardinalities of antichains in \mathbb{B} . Length {\rm length}(\mathbb{B}) of a Boolean algebra \mathbb{B} is {\rm length}(\mathbb{B}) = \sup\big\{|A| : A \subseteq \mathbb{B} \text{ is a chain} \big\} Depth {\rm depth}(\mathbb{B}) of a Boolean algebra \mathbb{B} is {\rm depth}(\mathbb{B}) = \sup\big\{|A| : A \subseteq \mathbb{B} \text{ is a well-ordered subset} \big\} . Incomparability {\rm Inc}(\mathbb{B}) of a Boolean algebra \mathbb{B} is {\rm Inc}({\mathbb B}) = \sup\big\{|A| : A \subseteq \mathbb{B} \text{ such that } \forall a,b \in A \big(a \neq b\ \Rightarrow \neg (a\leq b\ \vee \ b \leq a)\big)\big\} . Pseudo-weight \pi(\mathbb{B}) of a Boolean algebra \mathbb{B} is \pi(\mathbb{B}) = \min\big\{|A| : A \subseteq \mathbb{B}\setminus \{0\} \text{ such that } \forall b \in B\setminus\{0\} \big(\exists a \in A\big)\big(a \leq b\big)\big\}. ==Cardinal functions in algebra== Examples of cardinal functions in algebra are: Index of a subgroup H of G is the number of cosets. Dimension of a vector space V over a field K is the cardinality of any Hamel basis of V. More generally, for a free module M over a ring R we define rank {\rm rank}(M) as the cardinality of any basis of this module. For a linear subspace W of a vector space V we define codimension of W (with respect to V). For any algebraic structure it is possible to consider the minimal cardinality of generators of the structure. For algebraic field extensions, algebraic degree and separable degree are often employed (the algebraic degree equals the dimension of the extension as a vector space over the smaller field). For non-algebraic field extensions, transcendence degree is likewise used. ==External links== A Glossary of Definitions from General Topology [https://web.archive.org/web/20120204044336/https://math.berkeley.edu/~apollo/topodefs.ps ==See also== Cichoń's diagram

Text derived from https://en.wikipedia.org/w/index.php?oldid=1332585344 under CC BY-SA 4.0; modifications have been made.