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Wondering how to use Ultrafilter in a sentence? Below are 10+ example sentences from authentic English texts. Including the meaning .

Rare word

Ultrafilter in a sentence

Related words

Ultrafilter | Ultrafilters

Ultrafilter meaning

  1. A device that performs ultrafiltration.
  2. A proper filter which has a law of dichotomy for complements.
  3. A filter (subset of a poset) that is maximal as a set with respect to the definition of proper filter.

Using Ultrafilter

  • The main meaning on this page is: A device that performs ultrafiltration. | A proper filter which has a law of dichotomy for complements. | A filter (subset of a poset) that is maximal as a set with respect to the definition of proper filter.
  • In the example corpus, ultrafilter often appears in combinations such as: an ultrafilter, any ultrafilter, each ultrafilter.

Context around Ultrafilter

  • Average sentence length in these examples: 20.9 words
  • Position in the sentence: 5 start, 7 middle, 2 end
  • Sentence types: 14 statements, 0 questions, 0 exclamations

Often before:

principal 2× free 1×

Often after:

containing 2× consists 1× lemma 1× principal 1×

Sentence patterns with Ultrafilter

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

a principal ultrafilter consists of 1× an ultrafilter on ω 1× any ultrafilter containing a 1× any ultrafilter that is 1× called an ultrafilter because you 1× contains an ultrafilter has to 1× finite each ultrafilter is principal 1× for every ultrafilter f the 1×

Sentence openings and endings

Common sentence openings:

an ultrafilter on 1× any ultrafilter containing 1× any ultrafilter that 1× for ultrafilters on 1× gödel s ontological 1× one way to 1×

Common sentence endings:

is an ultrafilter 1× map is continuous 1× non principal ultrafilter 1× not the case 1× of this form 1× principal and free 1×

Words around Ultrafilter

Just before it in the word index:

ultrafast 22 ultrafan 2 ultraedge 3 ultraconservative 10 ultraconservatism 2 ultracompetitive 3

Just after it in the word index:

ultrafilters 11 ultrafiltration 5 ultrafine 5 ultragear 6 ultragenyx 12 ultrahand 13

Similar in corpus frequency:

aat 18 abms 18 abraxas 18 abrogating 18 absolutes 18 absorbance 18

Corpus analysis for Ultrafilter

  • In this selection, "ultrafilter" usually appears in the middle of the sentence. The average example has 20.9 words, and this corpus slice is mostly made up of statements.
  • Around the word, principal, free, containing, consists and lemma stand out and add context to how "ultrafilter" is used.
  • Recognizable usage signals include a principal ultrafilter consists of and an ultrafilter on ω. That gives this page its own corpus information beyond isolated example sentences.
  • By corpus frequency, "ultrafilter" sits close to words such as aat, abms and abraxas, which helps place it inside the broader word index.

Example types with ultrafilter

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

Any ultrafilter containing a finite set is trivial. (8 words)

If S is finite, each ultrafilter is principal. (8 words)

Any ultrafilter that is not principal is called a free (or non-principal) ultrafilter. (14 words)

Longer example sentences

One way to see that is by noting that the ultrafilter lemma (or equivalently, the Boolean prime ideal theorem), which is strictly weaker than the axiom of choice, can be used to show the Hahn–Banach theorem, although the converse is not the case. (44 words)

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 only explicitly known example of an ultrafilter is the family of sets containing a given element (in our case, say, the number 10). (24 words)

Example sentences (14)

Any ultrafilter that is not principal is called a free (or non-principal) ultrafilter.

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.

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

Any ultrafilter containing a finite set is trivial.

Gödel's ontological proof of God's existence uses as an axiom that the set of all "positive properties" is an ultrafilter.

If (1) also holds, U is called an ultrafilter (because you can add no more sets to it without breaking it).

If S is finite, each ultrafilter is principal.

One way to see that is by noting that the ultrafilter lemma (or equivalently, the Boolean prime ideal theorem), which is strictly weaker than the axiom of choice, can be used to show the Hahn–Banach theorem, although the converse is not the case.

Advertentie

The former is equivalent in ZF to the existence of an ultrafilter containing each given filter, proved by Tarski in 1930.

The only explicitly known example of an ultrafilter is the family of sets containing a given element (in our case, say, the number 10).

The operation is also right-continuous, in the sense that for every ultrafilter F, the map : : is continuous.

This implies that any filter that properly contains an ultrafilter has to be equal to the whole poset.

This is also notated A/U, directly in terms of the free ultrafilter U; the two are equivalent.

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

Advertentie

Common combinations with ultrafilter

These word pairs occur most frequently in English texts:

Frequently asked questions

How do you use "ultrafilter" in a sentence?
An example: "Any ultrafilter that is not principal is called a free (or non-principal) ultrafilter." This page contains 10+ example sentences with the word "ultrafilter" from authentic English texts.
What does "ultrafilter" mean?
Ultrafilter means: A device that performs ultrafiltration.
How many example sentences with "ultrafilter" are there?
Voorbeeldzinnen.info contains at least 10+ example sentences with "ultrafilter", drawn from a database of millions of English sentences.