See ultrafilter on Wiktionary
Download JSON data for ultrafilter meaning in All languages combined (5.4kB)
{
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"2": "ultra",
"3": "filter"
},
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],
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{
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{
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{
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"word": "free ultrafilter"
},
{
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"word": "principal ultrafilter"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
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{
"categories": [
{
"_dis": "23 17 28 32",
"kind": "other",
"name": "English terms prefixed with ultra-",
"parents": [],
"source": "w+disamb"
}
],
"glosses": [
"A device that performs ultrafiltration."
],
"id": "en-ultrafilter-en-noun-BUS2Hdok",
"links": [
[
"ultrafiltration",
"ultrafiltration"
]
]
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{
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{
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{
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"name": "English terms prefixed with ultra-",
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}
],
"examples": [
{
"text": "A proper filter of a set has these three properties: (1) it does not contain the empty set, (2) if it contains a subset then it contains all supersets of that subset, and (3) if it contains a pair of subsets then it also contains their intersection. To make it into an ultrafilter it must be made as large as possible without including the empty set. That can be prevented by not allowing any pair of disjoint sets to be both included. If, given a pair of complementary subsets, one of them is prevented from being included, then all subsets of it should be prevented from being included as well, by the second rule. That takes care of all subsets disjoint from the other complementary subset, which should then be included, in order to make the filter approach maximality, i.e., turn it into an ultrafilter."
}
],
"glosses": [
"A proper filter which has a law of dichotomy for complements."
],
"id": "en-ultrafilter-en-noun-zPr~hilf",
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],
"qualifier": "whose subsets are partially ordered by inclusion",
"raw_glosses": [
"(mathematics, of a set, whose subsets are partially ordered by inclusion) A proper filter which has a law of dichotomy for complements."
],
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"of a set"
],
"topics": [
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{
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{
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"examples": [
{
"text": "An ultrafilter is maximal in the sense that if any other element of the poset not already in it were added to it, one could deduce (from the laws which define the filter, and the given ordering relation, i.e., the structure of the poset) that the resulting filter must be improper; i.e., it must contain all the elements of the poset."
}
],
"glosses": [
"A filter (subset of a poset) that is maximal as a set with respect to the definition of proper filter."
],
"id": "en-ultrafilter-en-noun-Gl49C5n~",
"links": [
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"mathematics",
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],
"raw_glosses": [
"(mathematics, order theory) A filter (subset of a poset) that is maximal as a set with respect to the definition of proper filter."
],
"topics": [
"mathematics",
"order-theory",
"sciences"
],
"translations": [
{
"_dis1": "40 9 51",
"code": "fr",
"lang": "French",
"sense": "Translations",
"tags": [
"masculine"
],
"word": "ultrafiltre"
},
{
"_dis1": "40 9 51",
"code": "de",
"lang": "German",
"sense": "Translations",
"tags": [
"masculine"
],
"word": "Ultrafilter"
},
{
"_dis1": "40 9 51",
"code": "is",
"lang": "Icelandic",
"sense": "Translations",
"tags": [
"feminine"
],
"word": "ofursía"
},
{
"_dis1": "40 9 51",
"code": "pl",
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"sense": "Translations",
"tags": [
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"word": "ultrafiltr"
}
]
}
],
"word": "ultrafilter"
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{
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{
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],
"glosses": [
"To filter by ultrafiltration."
],
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"links": [
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"(transitive) To filter by ultrafiltration."
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{
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{
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"args": {},
"expansion": "ultrafilter (plural ultrafilters)",
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],
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{
"word": "free ultrafilter"
},
{
"word": "principal ultrafilter"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"glosses": [
"A device that performs ultrafiltration."
],
"links": [
[
"ultrafiltration",
"ultrafiltration"
]
]
},
{
"categories": [
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],
"examples": [
{
"text": "A proper filter of a set has these three properties: (1) it does not contain the empty set, (2) if it contains a subset then it contains all supersets of that subset, and (3) if it contains a pair of subsets then it also contains their intersection. To make it into an ultrafilter it must be made as large as possible without including the empty set. That can be prevented by not allowing any pair of disjoint sets to be both included. If, given a pair of complementary subsets, one of them is prevented from being included, then all subsets of it should be prevented from being included as well, by the second rule. That takes care of all subsets disjoint from the other complementary subset, which should then be included, in order to make the filter approach maximality, i.e., turn it into an ultrafilter."
}
],
"glosses": [
"A proper filter which has a law of dichotomy for complements."
],
"links": [
[
"mathematics",
"mathematics"
],
[
"filter",
"filter"
],
[
"dichotomy",
"dichotomy"
],
[
"complement",
"complement"
]
],
"qualifier": "whose subsets are partially ordered by inclusion",
"raw_glosses": [
"(mathematics, of a set, whose subsets are partially ordered by inclusion) A proper filter which has a law of dichotomy for complements."
],
"raw_tags": [
"of a set"
],
"topics": [
"mathematics",
"sciences"
]
},
{
"categories": [
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],
"examples": [
{
"text": "An ultrafilter is maximal in the sense that if any other element of the poset not already in it were added to it, one could deduce (from the laws which define the filter, and the given ordering relation, i.e., the structure of the poset) that the resulting filter must be improper; i.e., it must contain all the elements of the poset."
}
],
"glosses": [
"A filter (subset of a poset) that is maximal as a set with respect to the definition of proper filter."
],
"links": [
[
"mathematics",
"mathematics"
],
[
"filter",
"filter"
],
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"maximal"
],
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]
],
"raw_glosses": [
"(mathematics, order theory) A filter (subset of a poset) that is maximal as a set with respect to the definition of proper filter."
],
"topics": [
"mathematics",
"order-theory",
"sciences"
]
}
],
"translations": [
{
"code": "fr",
"lang": "French",
"sense": "Translations",
"tags": [
"masculine"
],
"word": "ultrafiltre"
},
{
"code": "de",
"lang": "German",
"sense": "Translations",
"tags": [
"masculine"
],
"word": "Ultrafilter"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "Translations",
"tags": [
"feminine"
],
"word": "ofursía"
},
{
"code": "pl",
"lang": "Polish",
"sense": "Translations",
"tags": [
"masculine"
],
"word": "ultrafiltr"
}
],
"word": "ultrafilter"
}
{
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"English nouns",
"English terms prefixed with ultra-",
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"lang": "English",
"lang_code": "en",
"pos": "verb",
"senses": [
{
"categories": [
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"glosses": [
"To filter by ultrafiltration."
],
"links": [
[
"ultrafiltration",
"ultrafiltration"
]
],
"raw_glosses": [
"(transitive) To filter by ultrafiltration."
],
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}
],
"word": "ultrafilter"
}
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-05-01 from the enwiktionary dump dated 2024-04-21 using wiktextract (f4fd8c9 and c9440ce). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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