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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.
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cofinality and partially
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.
* 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 ).
Thus the cofinality of a finite partially ordered set is equal to the number of its maximal elements.
cofinality and ordered
* A field F is complete if there is no ordered field K properly containing F such that F is dense in K. If the cofinality of K is κ, this is equivalent to saying Cauchy sequences indexed by κ are convergent in F.
PCF theory is the name of a mathematical theory, introduced by Saharon, that deals with the cofinality of the ultraproducts of ordered sets.
we let denote the cofinality of the ordered set of functions
cofinality and set
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.
** In particular, the cofinality of any nonzero finite ordinal, or indeed any finite directed set, is 1, since such sets have a greatest element.
This demonstrates that the cofinality depends on the order ; different orders on the same set may have different cofinality.
The cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set.
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality.
In other words, X is an unbounded sequence in F. The cofinality of F is the size of the smallest cofinal set, which is to say, the size of the smallest cardinality giving an unbounded sequence.
If is a cardinal of uncountable cofinality, and intersects every club set in then is called a stationary set.
cofinality and can
It can be seen to be the higher-dimensional analogue of the real numbers ; with cardinality instead of, cofinality instead of, and weight instead of, and with the η < sub > 1 </ sub > property in place of the η < sub > 0 </ sub > property ( which merely means between any two real numbers we can find another ).
cofinality and be
Some cardinal numbers cannot be proven to be equal to any particular aleph, for instance the cardinality of the continuum, whose value in ZFC may be any uncountable cardinal of uncountable cofinality ( see Easton's theorem ).
cofinality and least
The cofinality of any limit ordinal is at least ω.
cofinality and ordinal
The cofinality of an ordinal α is the smallest ordinal δ which is the order type of a cofinal subset of α.
For example, the cofinality of ω² is ω, because the sequence ω · m ( where m ranges over the natural numbers ) tends to ω² ; but, more generally, any countable limit ordinal has cofinality ω.
An uncountable limit ordinal may have either cofinality ω as does ω < sub > ω </ sub > or an uncountable cofinality.
The cofinality of any successor ordinal is 1.
A regular ordinal is an ordinal which is equal to its cofinality.
The cofinality of any ordinal α is a regular ordinal, i. e. the cofinality of the cofinality of α is the same as the cofinality of α.
For example, consider Namba forcing, that preserves and collapses to an ordinal of cofinality.
More generally, if the axiom of choice holds, it is provable that if there is a nontrivial elementary embedding of V < sub > α </ sub > into itself then α is either a limit ordinal of cofinality ω or the successor of such an ordinal.
cofinality and there
We are only forced to avoid setting it to certain special cardinals with cofinality, meaning there is an unbounded function from to it.
cofinality and is
Thus the cofinality of this poset is n choose m.
* A subset of the natural numbers N is cofinal in N if and only if it is infinite, and therefore the cofinality of ℵ < sub > 0 </ sub > is ℵ < sub > 0 </ sub >.
* The cofinality of the real numbers with their usual ordering is ℵ < sub > 0 </ sub >, since N is cofinal in R. The usual ordering of R is not order isomorphic to c, the cardinality of the real numbers, which has cofinality strictly greater than ℵ < sub > 0 </ sub >.
If two cofinal subsets of B have minimal cardinality ( i. e. their cardinality is the cofinality of B ), then they are order isomorphic to each other.
The cofinality of 0 is 0.
cofinality and function
where cf denotes the cofinality function ; the gimel function is used for studying the continuum function and the cardinal exponentiation function.
cofinality and cofinal
For example natural numbers are cofinal in the reals, and the cofinality of the reals is therefore.
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.
partially and ordered
** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.
** Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.
The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.
** Antichain principle: Every partially ordered set has a maximal antichain.
* Incidence algebras of locally finite partially ordered sets are unitary associative algebras considered in combinatorics.
The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set.
A partially ordered set ( poset ) P is said to satisfy the ascending chain condition ( ACC ) if every ascending chain of elements eventually terminates.
* Every cofinal subset of a partially ordered set must contain all maximal elements of that set.
This is partially ordered under inclusion and the subsets with m elements are maximal.
An abstract polyhedron is a certain kind of partially ordered set ( poset ) of elements, such that adjacencies, or connections, between elements of the set correspond to adjacencies between elements ( faces, edges, etc.
Directed sets are a generalization of nonempty totally ordered sets, that is, all totally ordered sets are directed sets ( contrast partially ordered sets which need not be directed ).
Presheaves: If X is a topological space, then the open sets in X form a partially ordered set Open ( X ) under inclusion.
Like every partially ordered set, Open ( X ) forms a small category by adding a single arrow U → V if and only if.
* In mathematics, a certain kind of subset of a partially ordered set.
** Filter ( mathematics ), a special subset of a partially ordered set
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