The isomorphism problem for some classes of multiplicative systems

Abstract

1. Introduction. We give here a solution of the isomorphism problem for finitely presented algebras in various classes of multiplicative systems. That is, we describe an algorithm for deciding whether two loops (quasigroups, groupoids, inverse property loops, etc.) given by finite sets of generators and defining relations are isomorphic. We give the details only for loops but with trivial changes, the procedure described here applies also to a number of other systems with a nonassociative multiplication which have been extensively discussed in the literature. The basic ideas used are those of an incomplete algebra [2 ; 4] and of a minimal incomplete algebra [6; 7]. With a finitely presented loop, we associate a finite minimal incomplete loop which freely generates the loop and we prove that two finitely presented loops are isomorphic if and only if the associated finite minimal incomplete loops are isomorphic. In other words, a finitely presented loop has (to within isomorphism) a unique finite minimal incomplete loop, which we call a basis of the loop, which freely generates the loop and, furthermore, a basis can be constructed in a finite number of steps from the generators and defining relations of the loop. Clearly, this solves the isomorphism problem for loops. In §2, we recall from [4] those properties of loops freely generated by incomplete loops needed in the rest of the paper. In §3, the crucial properties of minimal incomplete loops are derived and then used, in §4, to give a solution of the isomorphism property for loops. In the final section, we discuss the changes which have to be made to apply the ideas of this paper to the other multiplicative systems mentioned above. In addition, we obtain some results on the structure of finitely presented loops which follow from the properties of minimal incomplete loops. For example, a finitely presented loop which does not have a free loop as a free factor has only a finite number of automorphisms. Comparatively little is known about the isomorphism problem for various classes of algebras. It is solvable, of course, for finitely presented abelian groups and recently it has been shown to be unsolvable for finitely presented semigroups and groups. At the time the results of this paper were originally obtained [6], the author felt that similar methods could probably be applied to other varieties of algebras. In particular, it seemed plausible that the isomorphism problem could be solved, using the same ideas, for varieties satisfying the condition that incomplete