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A loop X is said to satisfy Moufang's theorem if for every x , y , z ∈ X such that x (y z) = (x y) z the subloop generated by x, y, z is a group. We prove that the variety V of Steiner loops satisfying the identity (x z) (((x y) z) (y z)) = ((x z) ((x y) z)) (y z) 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. [ABSTRACT FROM AUTHOR]

Möbius addition is defined on the complex open unit disk by and Möbius's exponential equation takes the form $$L(a\oplus _M b) = L(a)L(b)$$ , where L is a complex-valued function defined on the complex unit disk. In the present article, we indicate how Möbius's exponential equation is connected to Cauchy's exponential equation. Möbius's exponential equation arises when one determines the irreducible linear representations of the unit disk equipped with Möbius addition, considered as a nonassociative group-like structure. This suggests studying Schur's lemma in a more general setting. [ABSTRACT FROM AUTHOR]

We introduce a class of non-Moufang loops satisfying Moufang's theorem. [ABSTRACT FROM AUTHOR]