### Isotopy Invariants in Quasigroups

Transactions of the American Mathematical Society, Vol. 151, no. 2 (Oct. 1970), 511-526

Introduction

A major problem in the theory of loops is the isotopy-isomorphism problem—When does isotopy imply isomorphism? A closely related, more general problem is the isotopy-invariance problem-What properties are invariant under isotopy? The purpose of this paper is to investigate loop and quasi-group identities that are invariant under isotopy. The associative law and the Moufang identity (xy·x)z=x(y·xz) are well-known identities that are invariant with respect to loop isotopy. Since varieties are equational classes, an equivalent problem is the determination of all varieties of quasigroups and loops that are closed under isotopy.

§§3 to 6 are devoted to general considerations of the problem. It is shown 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 variety of loops. As a result we obtain a set of quasigroup identities such that every identity invariant under isotopy is either in this set or is equivalent to a member of the set. We next show that the varieties of quasigroups and the varieties of loops that are closed under isotopy form isomorphic lattices, L* and N*, respectively, with N* a sublattice of the lattice of loop varieties. An atom of L* that contains groups has the property that all of its free quasigroups of rank ℵ_{0} or less are isotopic. We also give a variety of loops, which contains no subvariety that is closed under isotopy except the trivial one.

The latter part of the paper is devoted to methods of generating varieties of loops that are isotopically closed. It is shown that the classes of central, left nuclear, right nuclear, and middle nuclear extensions of loops belonging to an isotopically closed loop variety are also isotopically closed varieties. In particular, for each integer n __>__ 1, the variety of centrally nilpotent loops of class m __<__ n is isotopically closed.
In addition, we show that any class of loops containing all loop isotopes of its members generates an isotopically closed loop variety. Because of the relationship between isotopically closed quasigroup and loop varieties we can generate an isotopically closed quasigroup variety from any of these new loop varieties by taking all isotopes of the members of the loop variety.