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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A special class of Moufang loops


Author: Hala Orlik-Pflugfelder
Journal: Proc. Amer. Math. Soc. 26 (1970), 583-586
MSC: Primary 20.95
DOI: https://doi.org/10.1090/S0002-9939-1970-0265498-1
MathSciNet review: 0265498
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Abstract: Loops satisfying the identical relation $ (xy)(zx) = [x(yz)]x$ are known as Moufang loops. In the present paper considered is a class of loops satisfying the identical relation

$\displaystyle (1)\qquad (xy)(z{x^\lambda }) = [x(yz)]{x^\lambda },$

where $ {x^\lambda }$ is the image of $ x$ under some mapping $ \lambda $ of the loop into itself. Loops satisfying (1) are called $ M$-loops.

If $ \lambda :x \to {x^k},k$ an integer, (1) is called an $ {M_k}$-law. It is shown that every $ M$-loop is Moufang, and every $ {x^\delta } = {x^{ - 1}} \cdot {x^\lambda }$ belongs to the nucleus.

Furthermore, if a loop satisfies (1) so do all its loop-isotopes.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0265498-1
Keywords: Moufang loop, inverse property loop (I.P. loop), di-associative property, neoring, $ M$-loop, $ {M_k}$-loop, strictly Moufang autotopism, pseudoautomorphism with companion $ C$, loop-isotopy
Article copyright: © Copyright 1970 American Mathematical Society