Isotopy is a generalization of the notion of isomorphism, appropriate for the classification of quasigroups and loops. A property is invariant under isotopy if, whenever it is satisfied by any quasigroup or loop G, it is also satisfied by every quasigroup or loop isotope of G. A major problem in quasigroup and loop theory is the determination of properties invariant under isotopy. The associative law and the Moufang identity are well known identities that are invariant with respect to loop isotopy. In this study, we determine a set of quasigroup identities that are invariant under isotopy and show that any other identity that is invariant under isotopy, is either in this set or equivalent to a member of this set.
Belousov has given a set of identities that are necessary and sufficient conditions that a quasigroup be isotopic to a group. We extend this set of identities and give other sets that are necessary and sufficient conditions that a quasigroup be isotopic to a M1, M2, or Moufang loop.
Varieties facilitate a study of the general case. We show that a variety of quasigroups is closed under isotopy if and only if it is the class of all quasigroups, all of whose loop isotopes lie in some isotopically closed variety of loops. The varieties of quasigroups and loops that are closed under isotopy form isomorphic lattices, Q* and L* with L*, a sublattice of the lattice of loop varieties. All free quasigroups is an atom of Q* that contains a group are isotopic.
Varieties of loops that are extensions of a normal subloop of their centers, left nuclei, middle nuclei or right nuclei, respectively, by a loop in an isotopically closed variety are also isotopically closed. In particular, for each n, the variety of nilpotent loops of class n is isotopically closed. Isotopically closed varieties can also be generated by any class of loops that contains all loop isotopes of its members. An example is given to show that not every variety contains a non-trivial variety closed under isotopy.
We consider the algebraic 3-nets of the members of an isotopically closed variety and show that any identity that is invariant under isotopy determines a closure condition for algebraic 3-nets that is equivalent to the identity.