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Loop near-rings and
unique decompositions of H-spaces
Damir Franetič and Petar Pavešić |
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Algebraic & Geometric Topology 16 (2016) 3563–3580
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Abstract |
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For every H-space , the set of homotopy classes possesses a natural algebraic structure of a loop near-ring. Albeit one cannot say much about general loop near-rings, it turns out that those that arise from H-spaces are sufficiently close to rings to have a viable Krull–Schmidt type decomposition theory, which is then reflected into decomposition results of H-spaces. In the paper, we develop the algebraic theory of local loop near-rings and derive an algebraic characterization of indecomposable and strongly indecomposable H-spaces. As a consequence, we obtain unique decomposition theorems for products of H-spaces. In particular, we are able to treat certain infinite products of H-spaces, thanks to a recent breakthrough in the Krull–Schmidt theory for infinite products. Finally, we show that indecomposable finite –local H-spaces are automatically strongly indecomposable, which leads to an easy alternative proof of classical unique decomposition theorems of Wilkerson and Gray. |
Keywords
H-space, near-ring, algebraic loop, idempotent, strongly
indecomposable space, Krull-Schmidt-Remak-Azumaya theorem
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Mathematical Subject Classification 2010
Primary: 55P45
Secondary: 16Y30
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References |
Publication
Received: 25 November 2015
Revised: 28 February 2016
Accepted: 20 May 2016
Published: 15 December 2016
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Authors |
| Damir Franetič | |
| Faculty of Computer and Information
Science University of Ljubljana Večna pot 113 1000 Ljubljana Slovenia |
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| http://www.fri.uni-lj.si/si/damir-franetic | |
| Petar Pavešić | |
| Faculty of Mathematics and
Physics University of Ljubljana Jadranska 19 1111 Ljubljana Slovenia |
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| http://www.fmf.uni-lj.si/~pavesic | |
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