**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Cofinality

Summary

In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A.
This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of a partially ordered set A can alternatively be defined as the least ordinal x such that there is a function from x to A with cofinal . This second definition makes sense without the axiom of choice. If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent.
Cofinality can be similarly defined for a directed set and is used to generalize the notion of a subsequence in a net.
The cofinality of a partially ordered set with greatest element is 1 as the set consisting only of the greatest element is cofinal (and must be contained in every other cofinal subset).
In particular, the cofinality of any nonzero finite ordinal, or indeed any finite directed set, is 1, since such sets have a greatest element.
Every cofinal subset of a partially ordered set must contain all maximal elements of that set. Thus the cofinality of a finite partially ordered set is equal to the number of its maximal elements.
In particular, let be a set of size and consider the set of subsets of containing no more than elements. This is partially ordered under inclusion and the subsets with elements are maximal. Thus the cofinality of this poset is choose
A subset of the natural numbers is cofinal in if and only if it is infinite, and therefore the cofinality of is Thus is a regular cardinal.
The cofinality of the real numbers with their usual ordering is since is cofinal in The usual ordering of is not order isomorphic to the cardinality of the real numbers, which has cofinality strictly greater than This demonstrates that the cofinality depends on the order; different orders on the same set may have different cofinality.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (2)

Related concepts (13)

Related lectures (3)

Aleph number

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (). The cardinality of the natural numbers is (read aleph-nought or aleph-zero; the term aleph-null is also sometimes used), the next larger cardinality of a well-ordered set is aleph-one then and so on.

Ordinal number

In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element").

Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers , sometimes called the continuum. It is an infinite cardinal number and is denoted by (lowercase Fraktur "c") or . The real numbers are more numerous than the natural numbers . Moreover, has the same number of elements as the power set of Symbolically, if the cardinality of is denoted as , the cardinality of the continuum is This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities.

Integers: Elementary Concepts

Covers fundamental concepts related to integers, including properties of well-ordered sets and the principle of induction.

Free Abelian Groups and Homomorphisms

Explores free abelian groups, homomorphisms, and exact sequences in the context of different categories.

Optimization Programs: Piecewise Linear Cost Functions

Covers the formulation of optimization programs for minimizing piecewise linear cost functions.

A set of phylogenetic trees with overlapping leaf sets is consistent if it can be merged without conflicts into a supertree. In this paper, we study the polynomial-time approximability of two related optimization problems called the maximum rooted triplets ...

Let X be a finite set and let k be a commutative ring. We consider the k-algebra of the monoid of all relations on X, modulo the ideal generated by the relations factorizing through a set of cardinality strictly smaller than Card(X), called inessential rel ...