See semigroup on Wiktionary
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(Specifically, the requirements for the existence of identity and inverse elements are omitted.)", "forms": [ { "form": "semigroups", "tags": [ "plural" ] } ], "head_templates": [ { "args": {}, "expansion": "semigroup (plural semigroups)", "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" ], "source": "w" }, { "kind": "other", "name": "English terms prefixed with semi-", "parents": [], "source": "w" }, { "kind": "other", "name": "Entries with translation boxes", "parents": [], "source": "w" }, { "kind": "other", "name": "Mandarin terms with non-redundant manual transliterations", "parents": [ "Terms with non-redundant manual transliterations", "Entry maintenance" ], "source": "w" }, { "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w" }, { "kind": "other", "name": "Pages with entries", "parents": [], "source": "w" }, { "kind": "other", "name": "Terms with Czech translations", "parents": [], "source": "w" }, { "kind": "other", "name": "Terms with Finnish translations", "parents": [], "source": "w" }, { "kind": "other", "name": "Terms with German translations", "parents": [], "source": "w" }, { "kind": "other", "name": "Terms with Icelandic translations", "parents": [], "source": "w" }, { "kind": "other", "name": "Terms with Italian translations", "parents": [], "source": "w" }, { "kind": "other", "name": "Terms with Mandarin translations", "parents": [], "source": "w" }, { "kind": "other", "name": "Terms with Portuguese translations", "parents": [], "source": "w" }, { "kind": "other", "name": "Terms with Romanian translations", "parents": [], "source": "w" }, { "kind": "other", "name": "Terms with Slovak translations", "parents": [], "source": "w" }, { "kind": "other", "name": "Terms with Swedish translations", "parents": [], "source": "w" }, { "kind": "topical", "langcode": "en", "name": "Mathematics", "orig": "en:Mathematics", "parents": [ "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" } ], "derived": [ { "word": "inverse semigroup" }, { "word": "semigroup homomorphism" }, { "word": "subsemigroup" } ], "examples": [ { "ref": "1961, Alfred Hoblitzelle Clifford, G. B. Preston, The Algebraic Theory of Semigroups, page 70:", "text": "If a semigroup S contains a zeroid, then every left zeroid is also a right zeroid, and vice versa, and the set K of all the zeroids of S is the kernel of S.", "type": "quote" }, { "text": "1988, A. Ya Aǐzenshtat, Boris M. Schein (translator), On Ideals of Semigroups of Endomorphisms, Ben Silver (editor), Nineteen Papers on Algebraic Semigroups, American Mathematical Society Translations, Series 2, Volume 139, page 11,\nIt follows naturally that various classes of ordered sets can be characterized by semigroup properties of endomorphism semigroups." }, { "ref": "2012, Jorge Almeida, Benjamin Steinberg, “Syntactic and Global Subgroup Theory: A Synthesis Approach”, in Jean-Camille Birget, Stuart Margolis, John Meakin, Mark V. Sapir, editors, Algorithmic Problems in Groups and Semigroups, page 5:", "text": "If one considers the variety of semigroups, one has the binary operation of multiplication defined on every semigroup.", "type": "quote" } ], "glosses": [ "Any set for which there is a binary operation that is closed and associative." ], "hypernyms": [ { "sense": "set for which a closed associative binary operation is defined", "word": "magma" } ], "hyponyms": [ { "sense": "set for which a closed associative binary operation is defined", "word": "group" }, { "sense": "set for which a closed associative binary operation is defined", "word": "monoid" } ], "id": "en-semigroup-en-noun-dQZ~vbtp", "links": [ [ "mathematics", "mathematics" ], [ "set", "set" ], [ "binary", "binary" ], [ "operation", "operation" ], [ "closed", "closed" ], [ "associative", "associative" ] ], "raw_glosses": [ "(mathematics) Any set for which there is a binary operation that is closed and associative." ], "topics": [ "mathematics", "sciences" ], "translations": [ { "code": "cmn", "lang": "Chinese Mandarin", "roman": "bàn qún", "sense": "set for which a closed associative binary operation is defined", "word": "半群" }, { "code": "cs", "lang": "Czech", "sense": "set for which a closed associative binary operation is defined", "tags": [ "feminine" ], "word": "pologrupa" }, { "code": "fi", "lang": "Finnish", "sense": "set for which a closed associative binary operation is defined", "word": "puoliryhmä" }, { "code": "de", "lang": "German", "sense": "set for which a closed associative binary operation is defined", "tags": [ "feminine" ], "word": "Halbgruppe" }, { "code": "is", "lang": "Icelandic", "sense": "set for which a closed associative binary operation is defined", "tags": [ "feminine" ], "word": "hálfgrúpa" }, { "code": "it", "lang": "Italian", "sense": "set for which a closed associative binary operation is defined", "tags": [ "masculine" ], "word": "semigruppo" }, { "code": "pt", "lang": "Portuguese", "sense": "set for which a closed associative binary operation is defined", "tags": [ "masculine" ], "word": "semigrupo" }, { "code": "ro", "lang": "Romanian", "sense": "set for which a closed associative binary operation is defined", "tags": [ "neuter" ], "word": "semigrup" }, { "code": "sk", "lang": "Slovak", "sense": "set for which a closed associative binary operation is defined", "tags": [ "feminine" ], "word": "pologrupa" }, { "code": "sv", "lang": "Swedish", "sense": "set for which a closed associative binary operation is defined", "word": "semigrupp" } ], "wikipedia": [ "semigroup" ] } ], "word": "semigroup" }
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