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On Steiner loops and power associativity. (English) Zbl 1131.20052
The author investigates Steiner loops that were introduced by {\it N. S. Mendelsohn} [Aequationes Math. 6, 228-230 (1971; Zbl 0244.20087)]. The author also provides six equivalent identities to characterize them by the Theorem: A groupoid $G(\cdot)$ is a generalized Steiner loop if and only if $G$ satisfies any one of the following identities: $a\cdot[((bb)\cdot c)\cdot a]=c$; $[a\cdot c(bb)]\cdot a=c$; $a\cdot (ca\cdot bb)=c$; $(a\cdot ca)\cdot bb=c$; $bb\cdot(a\cdot ca)=c$; $(bb\cdot a)\cdot(ca\cdot dd)=c$, for $a,b,c,d$ in $G$. The author proves the power associativity of Bol loops by using closure conditions. It is well known that left (right) Bol loops are power associative.
20N05Loops, quasigroups (group theory)
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