On Steiner loops and power associativity. (Q2461433)

The author investigates Steiner loops that were introduced by \textit{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.