Cofinal subset

Cofinal (mathematics)

In mathematics, a cofinal subset is a subset B of a preordered set A such that for every a in A there is a b in B such that ab. B is then said to be cofinal in A. Dually, a coinitial subset is a subset B of a preordered set A such that for every a in A there is a b in B such that a ≥ b. Usually, the preordered set is either a partially ordered set or a directed set.

A cofinal function is a function fX → A with preordered codomain A such that its range f(X) is cofinal in the codomain. A cofinal sequence is a sequence of elements of A such that its image is cofinal in A. A cofinal net is a net of elements of A such that its image is cofinal in A.

Concerning the cardinality of cofinal subsets, see cofinality.

Properties

Every partially ordered set is cofinal in itself. If B is a cofinal subset of a poset A and C is a cofinal subset of B with the partial ordering of A restricted to B, then C is also a cofinal subset of A. For a partially ordered set with maximal elements, every cofinal subset must contain all maximal elements. For a partially ordered set with greatest element, a subset is cofinal if and only if it contains that greatest element. Partially ordered sets without greatest element or maximal elements admit disjoint cofinal subsets. For example, the even and odd natural numbers form disjoint cofinal subsets of the set of all natural numbers.

If a partially ordered set A admits a totally ordered cofinal subset, then we can find a subset B which is well-ordered and cofinal in A.

Cofinal set of subsets

A particular but important case is given if A is a subset of the power set P(E) of some set E, ordered by reverse inclusion (⊃). Given this ordering of A, a subset B of A is cofinal in A if for every a ∈ A there is a b ∈ B such that a ⊃ b.

For example, if E is a group, A could be the set of normal subgroups of finite index. Then, cofinal subsets of A (or sequences, or nets) are used to define Cauchy sequences and the completion of the group.

See also

References

  • Lang, Serge (1997). Algebra (3rd ed., reprint w/ corr.). Addison-Wesley. ISBN 0-201-55540-9.

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