Abstract
There are many possible ways to define Moufang element. We show that the traditional definition is not the most felicitious — for instance, the set of all Moufang elements in an arbitrary loop, qua the traditional definition, need not form a subloop. We offer a new definition of Moufang element that ensures that the set of all Moufang elements in an arbitrary loop is a subloop. Moreover, this definition is "maximally algebraic" with respect to autotopisms. We also give an application of this new definition by showing that a flexible A-element in an inverse property loop is, in fact, a Moufang element, thus sharpening a well-known result of Kinyon, Kunen, and the present author [6]. Finally, we prove that divisible, Moufang groupoids are Moufang loops, thus sharpening a result of Kunen [9], one of the first computer-generated proofs in loop theory.
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