See poset on Wiktionary
Download JSON data for poset meaning in All languages combined (6.0kB)
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{
"text": "1973, Barbara L. Osofsky, Homological Dimensions of Modules, American Mathematical Society, →ISBN, page 76,\n42. Definition. A poset (partially ordered set) (X, ≤) (usually written just X) is a set X together with a transitive, antisymmetric relation ≤ on X.\n43. Definition. A linearly ordered set or chain is a poset (X, ≤), such that ∀a, b ∈ X, either a ≤ b or b ≤ a or a = b."
},
{
"text": "1998, Yuri A. Drozd, Representations of bisected posets and reflection functors, Idun Reiten, Sverre O. Smalø, Øyvind Solberg (editors), Algebras and Modules II, American Mathematical Society (for Canadian Mathematical Society), page 153,\nWe construct a complete set of reflection functors for the representations of posets and prove that they really have the usual properties. In particular, when the poset is of finite representation type, all of its indecomposable representations can be obtained from some \"trivial\" ones via relations. To define such reflection functors, a wider class of matrix problem is introduced, called \"representations of bisected posets\"."
},
{
"ref": "1999, Manfred Stern, Semimodular Lattices: Theory and Applications, Cambridge University Press, page 189",
"text": "The combinatorial interest in posets is largely due to two unoriented graphs associated with a given poset: the comparability graph (which we shall not consider here) and the covering graph.",
"type": "quotation"
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"id": "en-poset-sh-noun-8OaKwm3y",
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"tags": [
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"word": "pȍśet"
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}
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{
"text": "1973, Barbara L. Osofsky, Homological Dimensions of Modules, American Mathematical Society, →ISBN, page 76,\n42. Definition. A poset (partially ordered set) (X, ≤) (usually written just X) is a set X together with a transitive, antisymmetric relation ≤ on X.\n43. Definition. A linearly ordered set or chain is a poset (X, ≤), such that ∀a, b ∈ X, either a ≤ b or b ≤ a or a = b."
},
{
"text": "1998, Yuri A. Drozd, Representations of bisected posets and reflection functors, Idun Reiten, Sverre O. Smalø, Øyvind Solberg (editors), Algebras and Modules II, American Mathematical Society (for Canadian Mathematical Society), page 153,\nWe construct a complete set of reflection functors for the representations of posets and prove that they really have the usual properties. In particular, when the poset is of finite representation type, all of its indecomposable representations can be obtained from some \"trivial\" ones via relations. To define such reflection functors, a wider class of matrix problem is introduced, called \"representations of bisected posets\"."
},
{
"ref": "1999, Manfred Stern, Semimodular Lattices: Theory and Applications, Cambridge University Press, page 189",
"text": "The combinatorial interest in posets is largely due to two unoriented graphs associated with a given poset: the comparability graph (which we shall not consider here) and the covering graph.",
"type": "quotation"
}
],
"glosses": [
"A partially ordered set."
],
"links": [
[
"set theory",
"set theory"
],
[
"partially ordered set",
"partially ordered set"
]
],
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"(set theory, order theory) A partially ordered set."
],
"topics": [
"mathematics",
"order-theory",
"sciences",
"set-theory"
],
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"Partially ordered set"
]
}
],
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},
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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 2024-05-10 from the enwiktionary dump dated 2024-05-02 using wiktextract (a644e18 and edd475d). 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.