See identity element on Wiktionary
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"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": "quote"
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{
"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.",
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"An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other element."
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"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "additive identity"
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"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "multiplicative identity"
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"sense": "element that when applied with a binary operation leaves any other element unchanged",
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"word": "left identity"
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"word": "unity"
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"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "identity"
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"sense": "element that when applied with a binary operation leaves any other element unchanged",
"word": "neutral element"
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{
"code": "nl",
"lang": "Dutch",
"sense": "member of a structure",
"tags": [
"feminine"
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"word": "identiteit"
},
{
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"masculine"
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"word": "identitetselement"
}
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"word": "identity element"
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},
{
"ref": "2015, Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya. Subbotin, An Introduction to Essential Algebraic Structures, Wiley, page 41:",
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{
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"word": "egységelem"
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"tags": [
"masculine"
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"sense": "member of a structure",
"tags": [
"masculine"
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"code": "mi",
"lang": "Maori",
"sense": "member of a structure",
"word": "tūmau"
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{
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"word": "то́ждество"
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"code": "ru",
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"sense": "member of a structure",
"tags": [
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"word": "тожде́ственность"
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"tags": [
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"sense": "member of a structure",
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"word": "identitetselement"
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],
"word": "identity element"
}
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This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-11-06 from the enwiktionary dump dated 2024-10-02 using wiktextract (fbeafe8 and 7f03c9b). 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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