Automorphisms of finitary incidence rings
Let \(P\) be a quasiordered set, \(R\) an associative unital ring, \(\mathcal{C}(P,R)\) a partially ordered category associated with the pair \((P,R)\)[6], \(FI(P,R)\) a finitary incidence ring of \(\mathcal{C}(P,R)\)[6]. We prove that the group \({\rm Out}FI\) of outer automorphisms of \(FI(P,R)\)...
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Lugansk National Taras Shevchenko University
2018
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admjournalluguniveduua-article-6322018-04-04T09:11:25Z Automorphisms of finitary incidence rings Khripchenko, Nikolay finitary incidence algebra, partially ordered category,quasiordered set, automorphism 18E05, 18B35, 16S50, 16S60,16G20, 08A35 Let \(P\) be a quasiordered set, \(R\) an associative unital ring, \(\mathcal{C}(P,R)\) a partially ordered category associated with the pair \((P,R)\)[6], \(FI(P,R)\) a finitary incidence ring of \(\mathcal{C}(P,R)\)[6]. We prove that the group \({\rm Out}FI\) of outer automorphisms of \(FI(P,R)\) is isomorphic to the group \({\rm Out}\mathcal{C}\) of outer automorphisms of \(\mathcal{C}(P,R)\) under the assumption that \(R\) is indecomposable. In particular, if \(R\) is local, the equivalence classes of \(P\) are finite and \(P=\bigcup\limits_{i\in I}P_i\) is the decomposition of \(P\) into the disjoint union of the connected components, then \({\rm Out}FI\cong (H^1(\overline P,C(R)^*)\rtimes\prod\limits_{i\in I}{\rm Out}R)\rtimes{\rm Out}P\). Here \(H^1(\overline P,C(R)^*)\) is the first cohomology group of the order complex of the induced poset \(\overline P\) with the values in the multiplicative group of central invertible elements of \(R\). As a consequences, Theorem 2 [9], Theorem 5 [2 ] and Theorem 1.2 [8] are obtained. Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/632 Algebra and Discrete Mathematics; Vol 9, No 2 (2010) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/632/166 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
|
| datestamp_date |
2018-04-04T09:11:25Z |
| collection |
OJS |
| language |
English |
| topic |
finitary incidence algebra partially ordered category,quasiordered set automorphism 18E05 18B35 16S50 16S60,16G20 08A35 |
| spellingShingle |
finitary incidence algebra partially ordered category,quasiordered set automorphism 18E05 18B35 16S50 16S60,16G20 08A35 Khripchenko, Nikolay Automorphisms of finitary incidence rings |
| topic_facet |
finitary incidence algebra partially ordered category,quasiordered set automorphism 18E05 18B35 16S50 16S60,16G20 08A35 |
| format |
Article |
| author |
Khripchenko, Nikolay |
| author_facet |
Khripchenko, Nikolay |
| author_sort |
Khripchenko, Nikolay |
| title |
Automorphisms of finitary incidence rings |
| title_short |
Automorphisms of finitary incidence rings |
| title_full |
Automorphisms of finitary incidence rings |
| title_fullStr |
Automorphisms of finitary incidence rings |
| title_full_unstemmed |
Automorphisms of finitary incidence rings |
| title_sort |
automorphisms of finitary incidence rings |
| description |
Let \(P\) be a quasiordered set, \(R\) an associative unital ring, \(\mathcal{C}(P,R)\) a partially ordered category associated with the pair \((P,R)\)[6], \(FI(P,R)\) a finitary incidence ring of \(\mathcal{C}(P,R)\)[6]. We prove that the group \({\rm Out}FI\) of outer automorphisms of \(FI(P,R)\) is isomorphic to the group \({\rm Out}\mathcal{C}\) of outer automorphisms of \(\mathcal{C}(P,R)\) under the assumption that \(R\) is indecomposable. In particular, if \(R\) is local, the equivalence classes of \(P\) are finite and \(P=\bigcup\limits_{i\in I}P_i\) is the decomposition of \(P\) into the disjoint union of the connected components, then \({\rm Out}FI\cong (H^1(\overline P,C(R)^*)\rtimes\prod\limits_{i\in I}{\rm Out}R)\rtimes{\rm Out}P\). Here \(H^1(\overline P,C(R)^*)\) is the first cohomology group of the order complex of the induced poset \(\overline P\) with the values in the multiplicative group of central invertible elements of \(R\). As a consequences, Theorem 2 [9], Theorem 5 [2 ] and Theorem 1.2 [8] are obtained. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/632 |
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AT khripchenkonikolay automorphismsoffinitaryincidencerings |
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2025-12-02T15:40:30Z |
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2025-12-02T15:40:30Z |
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1850411613453025280 |