ⓘ Cardinal function. Cardinal characteristics of a proper ideal I of subsets of X are Aleph numbers and beth numbers can both be seen as cardinal functions define ..

                                     

ⓘ Cardinal function

  • Cardinal characteristics of a proper ideal I of subsets of X are
  • 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.
  • The most frequently used cardinal function is a function which assigns to a set "A" its cardinality, denoted by | A |.
a d I = min { | A |: A ⊆ I ∧ ⋃ A ∉ I }. {\displaystyle {\rm {add}}I=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}\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 ℵ 0 {\displaystyle \aleph _{0}} ; if I is a σ-ideal, then add ⁡ I ≥ ℵ 1. {\displaystyle \operatorname {add} I\geq \aleph _{1}.} cov ⁡ I = min { | A |: A ⊆ I ∧ ⋃ A = X }. {\displaystyle \operatorname {cov} I=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}=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 addI ≤ covI. non ⁡ I = min { | A |: A ⊆ X ∧ A ∉ I }, {\displaystyle \operatorname {non} I=\min\{|A|:A\subseteq X\ \wedge \ A\notin I{\big \}},} The "uniformity number" of I sometimes also written u n i f I {\displaystyle {\rm {unif}}I}) is the size of the smallest set not in I. Assuming I contains all singletons, addI ≤ nonI. c o f I = min { | B |: B ⊆ I ∧ ∀ A ∈ I ∃ B ∈ B A ⊆ B }. {\displaystyle {\rm {cof}}I=\min\{|{\mathcal {B}}|:{\mathcal {B}}\subseteq I\wedge \forall A\in I\exists B\in {\mathcal {B}}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 nonI ≤ cofI and covI ≤ cofI. In the case that I {\displaystyle 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 P, ⊑ {\displaystyle {\mathbb {P} },\sqsubseteq} the bounding number b P {\displaystyle {\mathfrak {b}}{\mathbb {P} }} and dominating number d P {\displaystyle {\mathfrak {d}}{\mathbb {P} }} is defined as
b P = min { | Y |: Y ⊆ P ∧ ∀ x ∈ P ∃ y ∈ y ⋢ x }, {\displaystyle {\mathfrak {b}}{\mathbb {P} }=\min {\big \{}|Y|:Y\subseteq {\mathbb {P} }\ \wedge \ \forall x\in {\mathbb {P} }\exists y\in Yy\not \sqsubseteq x{\big \}},} d P = min { | Y |: Y ⊆ P ∧ ∀ x ∈ P ∃ y ∈ Y x ⊑ y }. {\displaystyle {\mathfrak {d}}{\mathbb {P} }=\min {\big \{}|Y|:Y\subseteq {\mathbb {P} }\ \wedge \ \forall x\in {\mathbb {P} }\exists y\in Yx\sqsubseteq y{\big \}}.}
  • In PCF theory the cardinal function p κ λ {\displaystyle pp_{\kappa }\lambda} is used.
                                     

1. Cardinal functions in topology

Cardinal functions are widely used in topology as a tool for describing various topological properties. Below are some examples.

  • Perhaps the simplest cardinal invariants of a topological space X are its cardinality and the cardinality of its topology, denoted respectively by | X | and o X.
  • The π {\displaystyle \pi } -weight of a space X is the cardinality of the smallest π {\displaystyle \pi } -base for X.
  • The weight wX of a topological space X is the cardinality of the smallest base for X. When wX = ℵ 0 {\displaystyle \aleph _{0}} the space X is said to be second countable.
  • The network weight of X is the smallest cardinality of a network for X. A network is a family N {\displaystyle {\mathcal {N}}} of sets, for which, for all points x and open neighbourhoods U containing x, there exists B in N {\displaystyle {\mathcal {N}}} for which x ∈ B ⊆ U.
  • The density dX of a space X is the cardinality of the smallest dense subset of X. When d X = ℵ 0 {\displaystyle {\rm _{X}Z}.
  • The character of a topological space X at a point x is the cardinality of the smallest local base for x. The character of space X is χ X = sup { χ x, X: x ∈ X }. {\displaystyle \chi X=\sup \;\{\chi x,X:x\in X\}.} When χ X = ℵ 0 {\displaystyle \chi X=\aleph _{0}} the space X is said to be first countable.
                                     

2. 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 B {\displaystyle c{\mathbb {B} }} of a Boolean algebra B {\displaystyle {\mathbb {B} }} is the supremum of the cardinalities of antichains in B {\displaystyle {\mathbb {B} }}.
  • Length l e n g t h B {\displaystyle {\rm {length}}{\mathbb {B} }} of a Boolean algebra B {\displaystyle {\mathbb {B} }} is
l e n g t h B = sup { | A |: A ⊆ B {\displaystyle {\rm {length}}{\mathbb {B} }=\sup {\big \{}|A|:A\subseteq {\mathbb {B} }} is a chain } {\displaystyle {\big \}}}
  • Depth d e p t h B {\displaystyle {\rm {depth}}{\mathbb {B} }} of a Boolean algebra B {\displaystyle {\mathbb {B} }} is
d e p t h B = sup { | A |: A ⊆ B {\displaystyle {\rm {depth}}{\mathbb {B} }=\sup {\big \{}|A|:A\subseteq {\mathbb {B} }} is a well-ordered subset } {\displaystyle {\big \}}}.
  • Incomparability I n c B {\displaystyle {\rm {Inc}}{\mathbb {B} }} of a Boolean algebra B {\displaystyle {\mathbb {B} }} is
I n c B = sup { | A |: A ⊆ B {\displaystyle {\rm {Inc}}{\mathbb {B} }=\sup {\big \{}|A|:A\subseteq {\mathbb {B} }} such that ∀ a, b ∈ a ≠ b ⇒ ¬ a ≤ b ∨ b ≤ a) } {\displaystyle {\big }\forall a,b\in A{\big}{\big }a\neq b\ \Rightarrow \neg a\leq b\ \vee \ b\leq a{\big)}{\big \}}}.
  • Pseudo-weight π B {\displaystyle \pi {\mathbb {B} }} of a Boolean algebra B {\displaystyle {\mathbb {B} }} is
π B = min { | A |: A ⊆ B ∖ { 0 } {\displaystyle \pi {\mathbb {B} }=\min {\big \{}|A|:A\subseteq {\mathbb {B} }\setminus \{0\}} such that ∀ b ∈ B ∖ { 0 } ∃ a ∈ a ≤ b }. {\displaystyle {\big }\forall b\in B\setminus \{0\}{\big}{\big }\exists a\in A{\big}{\big }a\leq b{\big}{\big \}}.}
                                     

3. Cardinal functions in algebra

Examples of cardinal functions in algebra are:

  • For algebraic extensions algebraic degree and separable degree are often employed note that 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.
  • For any algebraic structure it is possible to consider the minimal cardinality of generators of the structure.
  • 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.
  • For a linear subspace W of a vector space V we define codimension of W with respect to V.
  • More generally, for a free module M over a ring R we define rank r a n k M {\displaystyle {\rm {rank}}M} as the cardinality of any basis of this module.
                                     
  • transfinite cardinal numbers describe the sizes of infinite sets. Cardinality is defined in terms of bijective functions Two sets have the same cardinality if
  • picture. A has cardinality less than or equal to the cardinality of B if there exists an injective function from A into B. A has cardinality strictly less
  • In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations
  • and cardinal indicators to characterize the belief. He also presented a lucid verbal and mathematical exposition of the social welfare function 1947
  • A cardinal Latin: Sanctae Romanae Ecclesiae cardinalis, literally cardinal of the Holy Roman Church is a leading bishop and prince of the College of
  • cardinality of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If f : X Y is a surjective function
  • In mathematics, an injective function also known as injection, or one - to - one function is a function that maps distinct elements of its domain to distinct
  • cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal κ
  • rational functions defined on the complex numbers, see Mobius transformation. The classical Mobius function μ n is an important multiplicative function in
  • approximately equivalent to the Cardinal Secretary of State, which absorbed its functions after the office of Cardinal Nephew was abolished in 1692. The
  • The Secretary of State of His Holiness The Pope, commonly known as the Cardinal Secretary of State, presides over the Holy See Secretariat of State, which