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Proceedings of the American Mathematical Society

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



Left loops which satisfy the left Bol identity

Author: B. L. Sharma
Journal: Proc. Amer. Math. Soc. 61 (1976), 189-195
MSC: Primary 20N05
MathSciNet review: 0422480
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Abstract: It is our purpose in this paper to initiate a study of the algebraic properties of a left loop $Q( \cdot )$ satisfying the identical relation \begin{equation} \tag {1} y(z \cdot yx) = (y \cdot zy)x \end{equation} for all $x,\;y,\;z \in Q$. It is shown that (1) implies right division in $Q( \cdot )$. By introducing a new operation ’$\circ$’ in $Q$, the connection between the left loop $Q( \cdot )$ and Bol loop $Q( \circ )$ is established. Further we show that the role of nuclei in the left loop theory is not the same as that in the loop theory. We conclude the paper by describing situations in which the left loop $Q( \cdot )$ is Moufang.

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Keywords: Left loop, Bol loop, Moufang loop, left (right) Bol property, nucleus, automorphism, left inverse property
Article copyright: © Copyright 1976 American Mathematical Society