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2016 Loop near-rings and unique decompositions of H-spaces
Damir Franetič, Petar Pavešić
Algebr. Geom. Topol. 16(6): 3563-3580 (2016). DOI: 10.2140/agt.2016.16.3563


For every H-space X, the set of homotopy classes [X,X] 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 p–local H-spaces are automatically strongly indecomposable, which leads to an easy alternative proof of classical unique decomposition theorems of Wilkerson and Gray.


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Damir Franetič. Petar Pavešić. "Loop near-rings and unique decompositions of H-spaces." Algebr. Geom. Topol. 16 (6) 3563 - 3580, 2016.


Received: 25 November 2015; Revised: 28 February 2016; Accepted: 20 May 2016; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1371.55006
MathSciNet: MR3584267
Digital Object Identifier: 10.2140/agt.2016.16.3563

Primary: 55P45
Secondary: 16Y30

Keywords: algebraic loop , H-space , idempotent , Krull-Schmidt-Remak-Azumaya theorem , Near-ring , strongly indecomposable space

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.16 • No. 6 • 2016
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