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.References
- G. Bol, Gewebe und gruppen, Math. Ann. 114 (1937), no. 1, 414–431 (German). MR 1513147, DOI 10.1007/BF01594185
- D. A. Robinson, Bol loops, Trans. Amer. Math. Soc. 123 (1966), 341–354. MR 194545, DOI 10.1090/S0002-9947-1966-0194545-4
- R. H. Bruck, Contributions to the theory of loops, Trans. Amer. Math. Soc. 60 (1946), 245–354. MR 17288, DOI 10.1090/S0002-9947-1946-0017288-3
- Richard Hubert Bruck, A survey of binary systems, Reihe: Gruppentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0093552
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