cofinalの例文

• Partially ordered sets without greatest element or maximal elements admit disjoint cofinal subsets.
• An ordinal is called regular if it is cofinal with any smaller ordinal; otherwise it is singular.
• For example, the second theorem above fails for the Tychonoff plank if we restrict ourselves to cofinal subnets.
• For example, the even and odd natural numbers form disjoint cofinal subsets of the set of all natural numbers.
• Cofinal subsets are very important in the theory of directed sets and cofinal subnet is the appropriate generalization of  subsequence.
• Cofinal subsets are very important in the theory of directed sets and cofinal subnet is the appropriate generalization of  subsequence.
• Every cyclically ordered group can be expressed as a quotient, where is a linearly ordered group and is a cyclic cofinal subgroup of.
• The cofinality of an ordinal \ alpha is the smallest ordinal \ delta that is the order type of a cofinal subset of \ alpha.
• A class of ordinal numbers is said to be unbounded, or cofinal, when given any ordinal, there is always some element of the class greater than it.
• In more formal language, the set of all left-hand Riemann sums and the set of all right-hand Riemann sums is cofinal in the set of all tagged partitions.
• 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 ).
• A more natural definition of a subnet would be to require " B " to be a cofinal subset of " A " and that " h " be the identity map.
• More generally, one can call a subset of any ordinal \ alpha cofinal in \ alpha provided every ordinal less than \ alpha is less than " or equal to " some ordinal in the set.
• 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 ".
• For a partially ordered set with maximal elements, every cofinal subset must contain all maximal elements, otherwise a maximal element which is not in the subset would fail to be " less than " any element of the subset, violating the definition of cofinal.
• For a partially ordered set with maximal elements, every cofinal subset must contain all maximal elements, otherwise a maximal element which is not in the subset would fail to be " less than " any element of the subset, violating the definition of cofinal.
• If V is a model of ZFC and its class of ordinals is regular, i . e . there is no cofinal subclass of lower order-type, then there is a closed unbounded class of ordinals, C, such that for every ? " C, the identity function from V ? to V is an elementary embedding.
• Rather than formulating these definitions for ( proper ) classes of ordinals, one can formulate them for sets of ordinals below a given ordinal \ alpha : A subset of a limit ordinal \ alpha is said to be unbounded ( or cofinal ) under \ alpha provided any ordinal less than \ alpha is less than some ordinal in the set.
• Of fundamental importance is the following theorem of Hausdorff : for each unbounded ordered dense set A there are two uniquely determined regular initial numbers \ omega _ { \ xi }, \ omega _ { \ eta } so that A is cofinal with \ omega _ { \ xi } and coinitial with \ omega _ { \ eta } ^ * ( * Denotes the inverse order ).
• Axiom I1 implies that V ? + 1 ( equivalently, H ( ? + ) ) does not satisfy V = HOD . There is no set S?" ? definable in V ? + 1 ( even from parameters V ? and ordinals < ? + ) with S cofinal in ? and | S | ? + 1 ) ( even from parameters in V ? ).