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Ultrafilters is an English word. Below you'll find 10+ example sentences showing how it's used in practice.

Rare word

Ultrafilters in a sentence

Related words

Ultrafilters | Ultrafilter

Ultrafilters meaning

plural of ultrafilter

Using Ultrafilters

  • The main meaning on this page is: plural of ultrafilter
  • In the example corpus, ultrafilters often appears in combinations such as: all ultrafilters, ultrafilters on, of ultrafilters.

Context around Ultrafilters

  • Average sentence length in these examples: 21.5 words
  • Position in the sentence: 5 start, 2 middle, 4 end
  • Sentence types: 11 statements, 0 questions, 0 exclamations

Often before:

using 1× powerset 1× free 1× uses 1×

Often after:

alternatively 1× turns 1×

Sentence patterns with Ultrafilters

These short patterns come directly from the example sentences and show how the word is often embedded:

1 of ultrafilters turns out 1× construction using ultrafilters alternatively if 1× existence of ultrafilters there are 1× for ultrafilters on a 1× of all ultrafilters of a 1× of all ultrafilters on x 1× of free ultrafilters 1× p point ultrafilters 1×

Sentence openings and endings

Common sentence openings:

an ultrafilter on 1× construction using ultrafilters 1× for ultrafilters on 1× proofs involving the 1× saharon shelah later 1× such ultrafilters are 1×

Common sentence endings:

as stone topology 1× element boolean algebra 1× extensions of structures 1× mentioned representation theorem 1× of free ultrafilters 1× of this form 1×

Words around Ultrafilters

Just before it in the word index:

ultrafilter 14 ultrafast 22 ultrafan 2 ultraedge 3 ultraconservative 10 ultraconservatism 2

Just after it in the word index:

ultrafiltration 5 ultrafine 5 ultragear 6 ultragenyx 12 ultrahand 13 ultrahigh 6

Similar in corpus frequency:

aab 14 aamer 14 aave 14 abadan 14 abdelmadjid 14 abdulkarim 14

Corpus analysis for Ultrafilters

  • In this selection, "ultrafilters" usually appears near the start of the sentence. The average example has 21.5 words, and this corpus slice is mostly made up of statements.
  • Around the word, using, powerset, free, alternatively and turns stand out and add context to how "ultrafilters" is used.
  • Recognizable usage signals include 1 of ultrafilters turns out and construction using ultrafilters alternatively if. That gives this page its own corpus information beyond isolated example sentences.
  • By corpus frequency, "ultrafilters" sits close to words such as aab, aamer and aave, which helps place it inside the broader word index.

Example types with ultrafilters

The same corpus examples are grouped by length and sentence type, making it easier to see the contexts in which the word appears:

Short example sentences

Proofs involving the axiom of choice do not produce explicit examples of free ultrafilters. (14 words)

The ultraproduct construction in model theory uses ultrafilters to produce elementary extensions of structures. (14 words)

Saharon Shelah later showed that it is consistent that there are no P-point ultrafilters. (15 words)

Longer example sentences

For ultrafilters on a powerset ℘(S), a principal ultrafilter consists of all subsets of S that contain a given element s of S. Each ultrafilter on ℘(S) that is also a principal filter is of this form. (37 words)

The set G of all ultrafilters of a poset P can be topologized in a natural way, that is in fact closely related to the above-mentioned representation theorem. (29 words)

Construction using ultrafilters Alternatively, if X is discrete, one can construct βX as the set of all ultrafilters on X, with a topology known as Stone topology. (27 words)

Example sentences (11)

Construction using ultrafilters Alternatively, if X is discrete, one can construct βX as the set of all ultrafilters on X, with a topology known as Stone topology.

An ultrafilter on ℘(ω) is Ramsey if and only if it is minimal in the Rudin–Keisler ordering of non-principal powerset ultrafilters.

For ultrafilters on a powerset ℘(S), a principal ultrafilter consists of all subsets of S that contain a given element s of S. Each ultrafilter on ℘(S) that is also a principal filter is of this form.

It can be proven by bisection method used in proving the Bolzano-Weierstrass theorem, the property (1) of ultrafilters turns out to be crucial.

Proofs involving the axiom of choice do not produce explicit examples of free ultrafilters.

Saharon Shelah later showed that it is consistent that there are no P-point ultrafilters.

Such ultrafilters are called trivial, and if we use it in our construction, we come back to the ordinary real numbers.

The set G of all ultrafilters of a poset P can be topologized in a natural way, that is in fact closely related to the above-mentioned representation theorem.

The ultraproduct construction in model theory uses ultrafilters to produce elementary extensions of structures.

Types and existence of ultrafilters There are two very different types of ultrafilter: principal and free.

Ultrafilters can alternatively be described as 2-valued morphisms from A to the two-element Boolean algebra.

Common combinations with ultrafilters

These word pairs occur most frequently in English texts:

Frequently asked questions

How do you use "ultrafilters" in a sentence?
An example: "Construction using ultrafilters Alternatively, if X is discrete, one can construct βX as the set of all ultrafilters on X, with a topology known as Stone topology." This page contains 10+ example sentences with the word "ultrafilters" from authentic English texts.
What does "ultrafilters" mean?
Ultrafilters means: plural of ultrafilter
How many example sentences with "ultrafilters" are there?
Voorbeeldzinnen.info contains at least 10+ example sentences with "ultrafilters", drawn from a database of millions of English sentences.