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On Steiner loops and power associativity. (English) Zbl 1131.20052
The author investigates Steiner loops that were introduced by 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 $G\left(·\right)$ is a generalized Steiner loop if and only if $G$ satisfies any one of the following identities: $a·\left[\left(\left(bb\right)·c\right)·a\right]=c$; $\left[a·c\left(bb\right)\right]·a=c$; $a·\left(ca·bb\right)=c$; $\left(a·ca\right)·bb=c$; $bb·\left(a·ca\right)=c$; $\left(bb·a\right)·\left(ca·dd\right)=c$, 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.
##### MSC:
 20N05 Loops, quasigroups (group theory)