Cofinal (mathematics)

In mathematics, a subset of a preordered set is said to be cofinal or frequent in if for every it is possible to find an element in that is "larger than " (explicitly, "larger than " means ). 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 is referred to as the cofinality of Let be a homogeneous binary relation on a set A subset is said to be or with respect to if it satisfies the following condition: For every there exists some that A subset that is not frequent is called . This definition is most commonly applied when is a directed set, which is a preordered set with additional properties. Final functions A map between two directed sets is said to be if the of is a cofinal subset of Coinitial subsets A subset is said to be (or in the sense of forcing) if it satisfies the following condition: For every there exists some such that This is the order-theoretic dual to the notion of cofinal subset. Cofinal (respectively coinitial) subsets are precisely the dense sets with respect to the right (respectively left) order topology. The cofinal relation over partially ordered sets ("posets") is reflexive: every poset is cofinal in itself. It is also transitive: if is a cofinal subset of a poset and is a cofinal subset of (with the partial ordering of applied to ), then is also a cofinal subset of For a partially ordered set with maximal elements, every cofinal subset must contain all maximal elements, otherwise a maximal element that is not in the subset would fail to be any element of the subset, violating the definition of cofinal. For a partially ordered set with a greatest element, a subset is cofinal if and only if it contains that greatest element (this follows, since a greatest element is necessarily a maximal element). Partially ordered sets without greatest element or maximal elements admit disjoint cofinal subsets.