Journal Title/Source

Journal of Algebra and its Applications

Publication Date

2009

Volume

8

Page Numbers

477–492

Document Type

Journal Article

Department

Mathematics and Computer Science

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 wellknown 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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