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Relatives of K-loops: Theory and examples. (English) Zbl 1038.20049
The paper presents various loop theoretic concepts, like the left inverse property, the automorphic inverse property, precession maps (generators of the left inner mapping group), nuclei, left power alternative elements and Bol’s law, and uses these concepts to define Kikkawa loops and K-loops, and to derive their basic properties. The concepts are defined, when applicable, for groupoids (with 1) and left loops, and their mutual dependencies are proved in this generalized setting. The notions defined are then examined with respect to a binary operation \(\circ\) defined on a group \(G\) by \(a\circ b=a\Psi_a(b)\), where \(\Psi_a\) is an automorphism of the group \(G\), for all \(a\in G\). Situations when \(\circ\) yields a loop give rise to a class of examples that show the independence of various properties. The examples are new; the preceding theory is basically a useful overview of already published material.

MSC:
20N05 Loops, quasigroups
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