The harmonious song of a Moufang quartet (Q6607376)

In this paper, the notion of Moufang loop is generalized to any magma. More specifically, the notion of Moufang magma, as a magma \((S,\cdot)\) with unit element satisfying the four well-known Moufang identities\N\begin{enumerate}\N\item[(A)] \(z(xy \cdot z) = zx \cdot yz\),\N\item[(B)] \((z \cdot xy)z = zx \cdot yz\),\N\item[(C)] \(z(x \cdot zy) = (zx \cdot z) y\), and\N\item[(D)] \((xz \cdot y)z = x(z \cdot yz)\)\N\end{enumerate}\Nfor all \(x,y,z\in S\) is introduced.\N\NIn addition, for each \(I\in \{\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\}\), it is defined the set \(I_\mu\) of elements \(a\in S\) satisfying the local version of the corresponding identity \((I)\) in which the variable \(z\) is replaced by \(a\). In particular, \(\mathrm{A}_\mu=\mathrm{B}_\mu\). Moreover, these four sets coincide whenever the magma under consideration satisfies the inverse property. The main result of this paper is the fact that a loop is Moufang if and only if all its elements belong to at least one of these four sets. That is, in order to prove that a loop is Moufang, it is not necessary to check the Moufang identities (A--D), but only to prove that each one of its elements satisfies any one of the four local identities. Some general properties satisfied by the set \(\mathrm{A}_\mu\cup \mathrm{B}_\mu \cup \mathrm{C}_\mu \cup \mathrm{D}_\mu\) are enumerated, with particular emphasis on those magmas with the right or left inverse property. The necessity of dealing with this last property is illustrated with some particular examples.