ABSTRACT
An inverse loop is a set with a distinguished element 1, binary operation xy, and inverse ⊣ satisfying x1=x=1xx⊣(xy)=y=(yx)x⊣(x⊣)⊣=x for all x, y. We define a monotopy to be any of the three maps in an isotopy. Hexads (D) and (E) show that we get the same concept whichever one we choose. Martin Liebeck has proved that the only finite simple Moufang loops other than groups are those consisting of the norm 1 elements in the octonions modulo p, taken modulo scalars. The chapter highlights the different forms of the Moufang Laws. The Moufang identities can be regarded as weakened forms of the associative law. One of the reasons for studying abstract Moufang loops is that these identities are equivalent to a symmetry property, namely that the monotopies are transitive.
