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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(·) is a generalized Steiner loop if and only if G satisfies any one of the following identities: a·[((bb)·c)·a]=c; [a·c(bb)]·a=c; a·(ca·bb)=c; (a·ca)·bb=c; bb·(a·ca)=c; (bb·a)·(ca·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.
MSC:
20N05Loops, quasigroups (group theory)