Finitary incidence algebras of quasiorders |
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| Author |
NSKhripchenko@mail.ru
Department of Mechanics and Mathematics, Kharkov V.~N.~Karazin National,University, 4 Svobody sq, 61077, Kharkov, Ukraine
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| Abstract |
We extend the notion of the finitary incidence algebra $FI(P,R)$ (see~[3]) to the case when $P(\preccurlyeq)$ is an arbitrary quasiordered set and $R$ is an associative unital ring. For each pair $(P,R)$ we build the preadditive category $\mathcal{C}(P,R)$ with $\overline P=P/_\sim$ as the set of objects. The isomorphism theorem for the finitary algebras is proved in the following weakened form: if $R$ and $S$ are the indecomposable rings, then $FI(P,R)\cong FI(Q,S)$ implies $\mathcal{C}(P,Q)\cong\mathcal{C}(Q,S)$. The results by Voss (Illinois J. Math., 1980, 24, 624--638) and the isomorphism theorem for weak incidence algebras (Int.J.Math. Math. Sci, 2004, 53, 2835--2845) are obtained as the consequences.
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| Keywords |
finitary incidence algebra, quasiordered set, associative unital ring, indecomposable ring, preadditive category
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| DOI |
doi:10.30970/ms.34.1.30-37
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Reference |
1. Abrams G., Haefner, J. and Á. del Río, Corrections and addenda to 'The isomorphism problem for incidence rings'// Pacific J. Math. -- 2002. -- V.207, No.2 -- P. 497--506.
2. Brown W.C. Matrices over Commutative Rings, Monographs and Textbooks in Pure and Applied Mathematics, V. 169. Marcel Dekker. New York, 1993. 3. Khripchenko N.S., Novikov B.V. Finitary incidence algebras// Communications in Algebra -- 2009. -- V.37, No.5. -- P. 1670--1676. 4. Singh S., Al-Thukair F. Weak incidence algebra and maximal ring of quotients// Int.J.Math. Math. Sci. -- 2004. -- No.53. -- P. 2835--2845. 5. Voss E.R. On the isomorphism problem for incidence rings// Illinois J. Math. -- 1980. -- No.24. -- P. 624--638. |
| Pages |
30-37
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| Volume |
34
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| Issue |
1
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| Year |
2010
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| Journal |
Matematychni Studii
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