See semisimple on Wiktionary
{
"etymology_templates": [
{
"args": {
"1": "en",
"2": "semi",
"3": "simple"
},
"expansion": "semi- + simple",
"name": "prefix"
}
],
"etymology_text": "From semi- + simple.",
"head_templates": [
{
"args": {
"1": "-"
},
"expansion": "semisimple (not comparable)",
"name": "en-adj"
}
],
"lang": "English",
"lang_code": "en",
"pos": "adj",
"senses": [
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Category theory",
"orig": "en:Category theory",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Mathematics",
"orig": "en:Mathematics",
"parents": [
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"_dis": "12 8 8 23 6 19 6 17",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w+disamb"
},
{
"_dis": "12 16 10 19 10 12 10 10",
"kind": "other",
"name": "English terms prefixed with semi-",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "11 7 7 24 6 26 6 15",
"kind": "other",
"name": "Pages with 2 entries",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "11 7 7 25 6 21 6 16",
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w+disamb"
}
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"Containing a collection of simple objects such that all objects in the category are direct sums of these simple objects."
],
"id": "en-semisimple-en-adj-sE3gqNOX",
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"category theory",
"category theory"
],
[
"abelian",
"abelian"
],
[
"category",
"category"
],
[
"simple",
"simple"
],
[
"object",
"object"
],
[
"direct sum",
"direct sum"
]
],
"qualifier": "most generally",
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(category theory, most generally, of an abelian category) Containing a collection of simple objects such that all objects in the category are direct sums of these simple objects."
],
"raw_tags": [
"of an abelian category",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"category-theory",
"computing",
"engineering",
"mathematics",
"natural-sciences",
"physical-sciences",
"sciences"
]
},
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Mathematics",
"orig": "en:Mathematics",
"parents": [
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"_dis": "12 16 10 19 10 12 10 10",
"kind": "other",
"name": "English terms prefixed with semi-",
"parents": [],
"source": "w+disamb"
}
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"In which each submodule is a direct summand; equivalently, equal to a direct sum of simple submodules."
],
"id": "en-semisimple-en-adj-e3LztbiN",
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"module",
"module"
],
[
"submodule",
"submodule"
],
[
"summand",
"summand"
],
[
"direct sum",
"direct sum"
]
],
"qualifier": "module theory",
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(module theory, of a module) In which each submodule is a direct summand; equivalently, equal to a direct sum of simple submodules."
],
"raw_tags": [
"of a module",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Mathematics",
"orig": "en:Mathematics",
"parents": [
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"_dis": "12 16 10 19 10 12 10 10",
"kind": "other",
"name": "English terms prefixed with semi-",
"parents": [],
"source": "w+disamb"
}
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple."
],
"id": "en-semisimple-en-adj-XOzku3QH",
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"ring",
"ring"
]
],
"qualifier": "ring theory",
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(ring theory, of an algebra or ring) Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple."
],
"raw_tags": [
"of an algebra or ring",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Mathematics",
"orig": "en:Mathematics",
"parents": [
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"_dis": "12 8 8 23 6 19 6 17",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w+disamb"
},
{
"_dis": "12 16 10 19 10 12 10 10",
"kind": "other",
"name": "English terms prefixed with semi-",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "11 7 7 24 6 26 6 15",
"kind": "other",
"name": "Pages with 2 entries",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "11 7 7 25 6 21 6 16",
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w+disamb"
}
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple.",
"Semiprimitive: having trivial Jacobson radical."
],
"id": "en-semisimple-en-adj-oMYjK525",
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"ring",
"ring"
],
[
"Semiprimitive",
"semiprimitive"
],
[
"Jacobson radical",
"Jacobson radical"
]
],
"qualifier": "ring theory",
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(ring theory, of an algebra or ring) Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple.",
"(of a ring, somewhat proscribed) Semiprimitive: having trivial Jacobson radical."
],
"raw_tags": [
"of a ring",
"of an algebraic structure",
"of an algebra or ring"
],
"synonyms": [
{
"word": "Jacobson semisimple"
}
],
"tags": [
"not-comparable",
"proscribed"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Linear algebra",
"orig": "en:Linear algebra",
"parents": [
"Algebra",
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Mathematics",
"orig": "en:Mathematics",
"parents": [
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"_dis": "12 16 10 19 10 12 10 10",
"kind": "other",
"name": "English terms prefixed with semi-",
"parents": [],
"source": "w+disamb"
}
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree."
],
"id": "en-semisimple-en-adj-wr-0xchB",
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"linear algebra",
"linear algebra"
],
[
"operator",
"operator"
],
[
"matrix",
"matrix"
],
[
"invariant",
"invariant"
],
[
"subspace",
"subspace"
],
[
"complement",
"complement"
],
[
"minimal",
"minimal"
],
[
"polynomial",
"polynomial"
],
[
"squarefree",
"squarefree"
]
],
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(linear algebra, of an operator or matrix) For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree."
],
"raw_tags": [
"of an operator or matrix",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"linear-algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Mathematics",
"orig": "en:Mathematics",
"parents": [
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"_dis": "12 8 8 23 6 19 6 17",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w+disamb"
},
{
"_dis": "12 16 10 19 10 12 10 10",
"kind": "other",
"name": "English terms prefixed with semi-",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "11 7 7 24 6 26 6 15",
"kind": "other",
"name": "Pages with 2 entries",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "11 7 7 25 6 21 6 16",
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w+disamb"
}
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"Being a direct sum of simple Lie algebras."
],
"id": "en-semisimple-en-adj-d3CM0FPq",
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"Lie algebra",
"Lie algebra"
],
[
"simple",
"simple"
]
],
"qualifier": "Lie theory",
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(Lie theory, of a Lie algebra) Being a direct sum of simple Lie algebras."
],
"raw_tags": [
"of a Lie algebra",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Mathematics",
"orig": "en:Mathematics",
"parents": [
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"_dis": "12 16 10 19 10 12 10 10",
"kind": "other",
"name": "English terms prefixed with semi-",
"parents": [],
"source": "w+disamb"
}
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"Being a direct sum of simple representations (also known as irreducible representations)."
],
"id": "en-semisimple-en-adj-fdlDJ66F",
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"linear representation",
"linear representation"
],
[
"group",
"group"
],
[
"simple representation",
"simple representation"
],
[
"irreducible representation",
"irreducible representation"
]
],
"qualifier": "representation theory",
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(representation theory, of a linear representation of a group or algebra) Being a direct sum of simple representations (also known as irreducible representations)."
],
"raw_tags": [
"of a linear representation of a group or algebra",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Group theory",
"orig": "en:Group theory",
"parents": [
"Algebra",
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Mathematics",
"orig": "en:Mathematics",
"parents": [
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"_dis": "12 8 8 23 6 19 6 17",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w+disamb"
},
{
"_dis": "12 16 10 19 10 12 10 10",
"kind": "other",
"name": "English terms prefixed with semi-",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "11 7 7 24 6 26 6 15",
"kind": "other",
"name": "Pages with 2 entries",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "11 7 7 25 6 21 6 16",
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w+disamb"
}
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"Being a linear algebraic group whose radical of the identity component is trivial."
],
"id": "en-semisimple-en-adj-eR4c48bR",
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"group theory",
"group theory"
],
[
"algebraic group",
"algebraic group"
],
[
"linear",
"linear"
],
[
"radical",
"radical"
],
[
"identity",
"identity"
],
[
"trivial",
"trivial"
]
],
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(group theory, of an algebraic group) Being a linear algebraic group whose radical of the identity component is trivial."
],
"raw_tags": [
"of an algebraic group",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"group-theory",
"mathematics",
"sciences"
]
}
],
"sounds": [
{
"audio": "LL-Q1860 (eng)-Flame, not lame-semisimple.wav",
"mp3_url": "https://upload.wikimedia.org/wikipedia/commons/transcoded/e/e2/LL-Q1860_%28eng%29-Flame%2C_not_lame-semisimple.wav/LL-Q1860_%28eng%29-Flame%2C_not_lame-semisimple.wav.mp3",
"ogg_url": "https://upload.wikimedia.org/wikipedia/commons/transcoded/e/e2/LL-Q1860_%28eng%29-Flame%2C_not_lame-semisimple.wav/LL-Q1860_%28eng%29-Flame%2C_not_lame-semisimple.wav.ogg"
}
],
"wikipedia": [
"semisimple"
],
"word": "semisimple"
}
{
"etymology_templates": [
{
"args": {
"1": "fr",
"2": "semi-",
"3": "simple"
},
"expansion": "semi- + simple",
"name": "af"
}
],
"etymology_text": "From semi- + simple.",
"forms": [
{
"form": "semisimples",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "semisimple (plural semisimples)",
"name": "fr-adj"
}
],
"lang": "French",
"lang_code": "fr",
"pos": "adj",
"senses": [
{
"categories": [
{
"kind": "other",
"name": "French entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "French terms prefixed with semi-",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with 2 entries",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w"
},
{
"kind": "topical",
"langcode": "fr",
"name": "Mathematics",
"orig": "fr:Mathematics",
"parents": [
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
}
],
"glosses": [
"semisimple"
],
"id": "en-semisimple-fr-adj-fCG47oTO",
"links": [
[
"mathematics",
"mathematics"
],
[
"semisimple",
"semisimple#English"
]
],
"raw_glosses": [
"(mathematics) semisimple"
],
"synonyms": [
{
"word": "semi-simple"
}
],
"topics": [
"mathematics",
"sciences"
]
}
],
"word": "semisimple"
}
{
"categories": [
"English adjectives",
"English entries with incorrect language header",
"English lemmas",
"English terms prefixed with semi-",
"English uncomparable adjectives",
"Pages with 2 entries",
"Pages with entries"
],
"etymology_templates": [
{
"args": {
"1": "en",
"2": "semi",
"3": "simple"
},
"expansion": "semi- + simple",
"name": "prefix"
}
],
"etymology_text": "From semi- + simple.",
"head_templates": [
{
"args": {
"1": "-"
},
"expansion": "semisimple (not comparable)",
"name": "en-adj"
}
],
"lang": "English",
"lang_code": "en",
"pos": "adj",
"senses": [
{
"categories": [
"en:Algebra",
"en:Category theory",
"en:Mathematics"
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"Containing a collection of simple objects such that all objects in the category are direct sums of these simple objects."
],
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"category theory",
"category theory"
],
[
"abelian",
"abelian"
],
[
"category",
"category"
],
[
"simple",
"simple"
],
[
"object",
"object"
],
[
"direct sum",
"direct sum"
]
],
"qualifier": "most generally",
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(category theory, most generally, of an abelian category) Containing a collection of simple objects such that all objects in the category are direct sums of these simple objects."
],
"raw_tags": [
"of an abelian category",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"category-theory",
"computing",
"engineering",
"mathematics",
"natural-sciences",
"physical-sciences",
"sciences"
]
},
{
"categories": [
"en:Algebra",
"en:Mathematics"
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"In which each submodule is a direct summand; equivalently, equal to a direct sum of simple submodules."
],
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"module",
"module"
],
[
"submodule",
"submodule"
],
[
"summand",
"summand"
],
[
"direct sum",
"direct sum"
]
],
"qualifier": "module theory",
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(module theory, of a module) In which each submodule is a direct summand; equivalently, equal to a direct sum of simple submodules."
],
"raw_tags": [
"of a module",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
"en:Algebra",
"en:Mathematics"
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple."
],
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"ring",
"ring"
]
],
"qualifier": "ring theory",
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(ring theory, of an algebra or ring) Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple."
],
"raw_tags": [
"of an algebra or ring",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
"en:Algebra",
"en:Mathematics"
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple.",
"Semiprimitive: having trivial Jacobson radical."
],
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"ring",
"ring"
],
[
"Semiprimitive",
"semiprimitive"
],
[
"Jacobson radical",
"Jacobson radical"
]
],
"qualifier": "ring theory",
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(ring theory, of an algebra or ring) Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple.",
"(of a ring, somewhat proscribed) Semiprimitive: having trivial Jacobson radical."
],
"raw_tags": [
"of a ring",
"of an algebraic structure",
"of an algebra or ring"
],
"synonyms": [
{
"word": "Jacobson semisimple"
}
],
"tags": [
"not-comparable",
"proscribed"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
"en:Algebra",
"en:Linear algebra",
"en:Mathematics"
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree."
],
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"linear algebra",
"linear algebra"
],
[
"operator",
"operator"
],
[
"matrix",
"matrix"
],
[
"invariant",
"invariant"
],
[
"subspace",
"subspace"
],
[
"complement",
"complement"
],
[
"minimal",
"minimal"
],
[
"polynomial",
"polynomial"
],
[
"squarefree",
"squarefree"
]
],
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(linear algebra, of an operator or matrix) For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree."
],
"raw_tags": [
"of an operator or matrix",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"linear-algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
"en:Algebra",
"en:Mathematics"
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"Being a direct sum of simple Lie algebras."
],
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"Lie algebra",
"Lie algebra"
],
[
"simple",
"simple"
]
],
"qualifier": "Lie theory",
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(Lie theory, of a Lie algebra) Being a direct sum of simple Lie algebras."
],
"raw_tags": [
"of a Lie algebra",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
"en:Algebra",
"en:Mathematics"
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"Being a direct sum of simple representations (also known as irreducible representations)."
],
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"linear representation",
"linear representation"
],
[
"group",
"group"
],
[
"simple representation",
"simple representation"
],
[
"irreducible representation",
"irreducible representation"
]
],
"qualifier": "representation theory",
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(representation theory, of a linear representation of a group or algebra) Being a direct sum of simple representations (also known as irreducible representations)."
],
"raw_tags": [
"of a linear representation of a group or algebra",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
"en:Algebra",
"en:Group theory",
"en:Mathematics"
],
"glosses": [
"In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"Being a linear algebraic group whose radical of the identity component is trivial."
],
"links": [
[
"mathematics",
"mathematics"
],
[
"algebra",
"algebra"
],
[
"algebraic structure",
"algebraic structure"
],
[
"group theory",
"group theory"
],
[
"algebraic group",
"algebraic group"
],
[
"linear",
"linear"
],
[
"radical",
"radical"
],
[
"identity",
"identity"
],
[
"trivial",
"trivial"
]
],
"raw_glosses": [
"(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.",
"(group theory, of an algebraic group) Being a linear algebraic group whose radical of the identity component is trivial."
],
"raw_tags": [
"of an algebraic group",
"of an algebraic structure"
],
"tags": [
"not-comparable"
],
"topics": [
"algebra",
"group-theory",
"mathematics",
"sciences"
]
}
],
"sounds": [
{
"audio": "LL-Q1860 (eng)-Flame, not lame-semisimple.wav",
"mp3_url": "https://upload.wikimedia.org/wikipedia/commons/transcoded/e/e2/LL-Q1860_%28eng%29-Flame%2C_not_lame-semisimple.wav/LL-Q1860_%28eng%29-Flame%2C_not_lame-semisimple.wav.mp3",
"ogg_url": "https://upload.wikimedia.org/wikipedia/commons/transcoded/e/e2/LL-Q1860_%28eng%29-Flame%2C_not_lame-semisimple.wav/LL-Q1860_%28eng%29-Flame%2C_not_lame-semisimple.wav.ogg"
}
],
"wikipedia": [
"semisimple"
],
"word": "semisimple"
}
{
"etymology_templates": [
{
"args": {
"1": "fr",
"2": "semi-",
"3": "simple"
},
"expansion": "semi- + simple",
"name": "af"
}
],
"etymology_text": "From semi- + simple.",
"forms": [
{
"form": "semisimples",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "semisimple (plural semisimples)",
"name": "fr-adj"
}
],
"lang": "French",
"lang_code": "fr",
"pos": "adj",
"senses": [
{
"categories": [
"French adjectives",
"French entries with incorrect language header",
"French lemmas",
"French terms prefixed with semi-",
"Pages with 2 entries",
"Pages with entries",
"fr:Mathematics"
],
"glosses": [
"semisimple"
],
"links": [
[
"mathematics",
"mathematics"
],
[
"semisimple",
"semisimple#English"
]
],
"raw_glosses": [
"(mathematics) semisimple"
],
"topics": [
"mathematics",
"sciences"
]
}
],
"synonyms": [
{
"word": "semi-simple"
}
],
"word": "semisimple"
}
Download raw JSONL data for semisimple meaning in All languages combined (9.9kB)
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2025-01-08 from the enwiktionary dump dated 2025-01-01 using wiktextract (9a96ef4 and 4ed51a5). 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.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.