A variety of Steiner loops satisfying Moufang’s theorem: a solution to Rajah’s Problem
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A loop X is said to satisfy Moufang’s theorem if for every \(x,y,z\in X\) such that \(x(yz)=(xy)z\) the subloop generated by x, y, z is a group. We prove that the variety V of Steiner loops satisfying the identity \((xz)(((xy)z)(yz)) = ((xz)((xy)z))(yz)\) is not contained in the variety of Moufang loops, yet every loop in V satisfies Moufang’s theorem. This solves a problem posed by Andrew Rajah.
KeywordsMoufang’s theorem Steiner loop Steiner triple system Pasch configuration
Mathematics Subject Classification20N05 05B07
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