See identity element in All languages combined, or Wiktionary
Download JSON data for identity element meaning in English (4.3kB)
{
"forms": [
{
"form": "identity elements",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "identity element (plural identity elements)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
{
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
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"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
}
],
"examples": [
{
"ref": "1990, Daniel M. Fendel, Diane Resek, Foundations of Higher Mathematics, Volume 1, Addison-Wesley, page 269",
"text": "Therefore the number 0 is not considered an identity element for subtraction, even though x-0#x3D;x for all x, since 0-x#x5C;nex.",
"type": "quotation"
},
{
"ref": "2003, Houshang H. Sohrab, Basic Real Analysis, Birkhäuser, page 17, Let (G,·) be a group. Then the identity element e∈G is unique. […]",
"text": "Proof. If e and e' are both identity elements, then we have ee'=e since e' is an identity element, and ee'=e' since e is an identity element. Thus\ne=ee'=e'."
},
{
"ref": "2015, Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya. Subbotin, An Introduction to Essential Algebraic Structures, Wiley, page 41",
"text": "Sometimes, to avoid ambiguity, we may use the notation e#x5F;M for the identity element of M.\nIf multiplicative notation is used then we use the term identity element, and often use the notation 1, or 1#x5F;M, for the neutral element e.",
"type": "quotation"
}
],
"glosses": [
"An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other element."
],
"hyponyms": [
{
"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "additive identity"
},
{
"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "multiplicative identity"
},
{
"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "zero"
},
{
"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "zero element"
}
],
"id": "en-identity_element-en-noun-cOZYX~Ty",
"links": [
[
"algebra",
"algebra"
],
[
"binary operation",
"binary operation"
]
],
"raw_glosses": [
"(algebra) An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other element."
],
"related": [
{
"word": "left identity"
},
{
"word": "right identity"
},
{
"word": "unity"
}
],
"synonyms": [
{
"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "identity"
},
{
"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "neutral element"
}
],
"topics": [
"algebra",
"mathematics",
"sciences"
],
"translations": [
{
"code": "nl",
"lang": "Dutch",
"sense": "member of a structure",
"tags": [
"feminine"
],
"word": "identiteit"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "member of a structure",
"word": "identiteettialkio"
},
{
"code": "he",
"lang": "Hebrew",
"sense": "member of a structure",
"tags": [
"masculine"
],
"word": "איבר יחידה"
},
{
"code": "hu",
"lang": "Hungarian",
"sense": "member of a structure",
"word": "egységelem"
},
{
"code": "ga",
"lang": "Irish",
"sense": "member of a structure",
"tags": [
"masculine"
],
"word": "ball ionannais"
},
{
"code": "it",
"lang": "Italian",
"sense": "member of a structure",
"tags": [
"masculine"
],
"word": "elemento neutro"
},
{
"code": "mi",
"lang": "Maori",
"sense": "member of a structure",
"word": "tūmau"
},
{
"code": "ru",
"lang": "Russian",
"roman": "tóždestvo",
"sense": "member of a structure",
"tags": [
"neuter"
],
"word": "то́ждество"
},
{
"code": "ru",
"lang": "Russian",
"roman": "toždéstvennostʹ",
"sense": "member of a structure",
"tags": [
"feminine"
],
"word": "тожде́ственность"
},
{
"code": "es",
"lang": "Spanish",
"sense": "member of a structure",
"tags": [
"masculine"
],
"word": "elemento neutro"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "member of a structure",
"tags": [
"neuter"
],
"word": "identitetselement"
}
],
"wikipedia": [
"identity element"
]
}
],
"word": "identity element"
}
{
"forms": [
{
"form": "identity elements",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "identity element (plural identity elements)",
"name": "en-noun"
}
],
"hyponyms": [
{
"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "additive identity"
},
{
"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "multiplicative identity"
},
{
"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "zero"
},
{
"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "zero element"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "left identity"
},
{
"word": "right identity"
},
{
"word": "unity"
}
],
"senses": [
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English lemmas",
"English multiword terms",
"English nouns",
"English terms with quotations",
"en:Algebra"
],
"examples": [
{
"ref": "1990, Daniel M. Fendel, Diane Resek, Foundations of Higher Mathematics, Volume 1, Addison-Wesley, page 269",
"text": "Therefore the number 0 is not considered an identity element for subtraction, even though x-0#x3D;x for all x, since 0-x#x5C;nex.",
"type": "quotation"
},
{
"ref": "2003, Houshang H. Sohrab, Basic Real Analysis, Birkhäuser, page 17, Let (G,·) be a group. Then the identity element e∈G is unique. […]",
"text": "Proof. If e and e' are both identity elements, then we have ee'=e since e' is an identity element, and ee'=e' since e is an identity element. Thus\ne=ee'=e'."
},
{
"ref": "2015, Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya. Subbotin, An Introduction to Essential Algebraic Structures, Wiley, page 41",
"text": "Sometimes, to avoid ambiguity, we may use the notation e#x5F;M for the identity element of M.\nIf multiplicative notation is used then we use the term identity element, and often use the notation 1, or 1#x5F;M, for the neutral element e.",
"type": "quotation"
}
],
"glosses": [
"An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other element."
],
"links": [
[
"algebra",
"algebra"
],
[
"binary operation",
"binary operation"
]
],
"raw_glosses": [
"(algebra) An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other element."
],
"topics": [
"algebra",
"mathematics",
"sciences"
],
"wikipedia": [
"identity element"
]
}
],
"synonyms": [
{
"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "identity"
},
{
"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "neutral element"
}
],
"translations": [
{
"code": "nl",
"lang": "Dutch",
"sense": "member of a structure",
"tags": [
"feminine"
],
"word": "identiteit"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "member of a structure",
"word": "identiteettialkio"
},
{
"code": "he",
"lang": "Hebrew",
"sense": "member of a structure",
"tags": [
"masculine"
],
"word": "איבר יחידה"
},
{
"code": "hu",
"lang": "Hungarian",
"sense": "member of a structure",
"word": "egységelem"
},
{
"code": "ga",
"lang": "Irish",
"sense": "member of a structure",
"tags": [
"masculine"
],
"word": "ball ionannais"
},
{
"code": "it",
"lang": "Italian",
"sense": "member of a structure",
"tags": [
"masculine"
],
"word": "elemento neutro"
},
{
"code": "mi",
"lang": "Maori",
"sense": "member of a structure",
"word": "tūmau"
},
{
"code": "ru",
"lang": "Russian",
"roman": "tóždestvo",
"sense": "member of a structure",
"tags": [
"neuter"
],
"word": "то́ждество"
},
{
"code": "ru",
"lang": "Russian",
"roman": "toždéstvennostʹ",
"sense": "member of a structure",
"tags": [
"feminine"
],
"word": "тожде́ственность"
},
{
"code": "es",
"lang": "Spanish",
"sense": "member of a structure",
"tags": [
"masculine"
],
"word": "elemento neutro"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "member of a structure",
"tags": [
"neuter"
],
"word": "identitetselement"
}
],
"word": "identity element"
}
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-03 from the enwiktionary dump dated 2024-05-02 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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