Moufang Loops, Magmas And The Moufang Identities - Zaku, Garba Gambo (2020)
Loop theory is a generalization of group theory; Moufang loops are a variety of loops. Four equivalent (Moufang) identities axiomatize these loops. Moufang loops also share many similar properties as groups though generally they are not associative; Moufang’s Theorem is pivotal in establishing this close relationship. The existing proof of the equivalence of the Moufang identities involves the notion of "autotopism", a completely difficult concept in itself, whereas there is no known complete proof of the Moufang’s Theorem (though several reasonably acceptable proofs exist). This thesis provides a simple, basic and complete proof of both. Moreover, the equivalence of the localized versions of the four identities is studied under the generalized setting of magmas and proven under necessary and sufficient conditions. Finally, this research gives a (partial) resolution of Moufang loops of odd order p2q4.