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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Left loops which satisfy the left Bol identity
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by B. L. Sharma PDF
Proc. Amer. Math. Soc. 61 (1976), 189-195 Request permission

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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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 61 (1976), 189-195
  • MSC: Primary 20N05
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0422480-4
  • MathSciNet review: 0422480