Cofinal Subgroup
Cofinal may refer to: * Cofinal (mathematics) * Cofinality (mathematics) *Cofinal (music) A Gregorian mode (or church mode) is one of the eight systems of pitch organization used in Gregorian chant. History The name of Pope Gregory I was attached to the variety of chant that was to become the dominant variety in medieval western and ...

Cofinal (mathematics)
In mathematics, a subset B \subseteq A of a preordered set (A, \leq) is said to be cofinal or frequent in A if for every a \in A, it is possible to find an element b in B that is "larger than a" (explicitly, "larger than a" means a \leq b). Cofinal subsets are very important in the theory of directed sets and nets, where “ cofinal subnet” is the appropriate generalization of "subsequence". They are also important in order theory, including the theory of cardinal numbers, where the minimum possible cardinality of a cofinal subset of A is referred to as the cofinality of A. Definitions Let \,\leq\, be a homogeneous binary relation on a set A. A subset B \subseteq A is said to be or with respect to \,\leq\, if it satisfies the following condition: :For every a \in A, there exists some b \in B that a \leq b. A subset that is not frequent is called . This definition is most commonly applied when (A, \leq) is a directed set, which is a preordered set with additiona ...
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Cofinality
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 image. 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. Examples * 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 subse ...
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