In a recent paper, those quasigroup identities involving at most three variables and of “length” six which force the quasigroup to be a loop or group have been enumerated by computer. We separate these identities into subsets according to what classes of loops they define and also provide humanly-comprehensible proofs for most of the computer-generated results.

In this paper, we shall study type-definable groups in a simple theory with respect to one or several stable reducts. While the original motivation came from the analysis of definable groups in structures obtained by Hrushovski's amalgamation method, the notions introduced are in fact more general, and in particular can be applied to certain expansions of algebraically closed fields by operators.

This is a survey of the results obtained by K. Głazek and his co-workers. We restrict our attention to the problems of axiomatizations of n-ary groups, classes of n-ary groups, properties of skew elements and homomorphisms induced by skew elements, constructions of covering groups, classifications and representations of n-ary groups. Some new results are added too.

If $(G,·)$ is a group, and the operation $(*)$ is defined by $x*y=x·y^{-1}$ then by direct verification $(G,*)$ is a quasigroup which satisfies the identity $(x*y)*(z*y)=x*z$. Conversely, if one starts with a quasigroup satisfying the latter identity the group $(G,·)$ can be constructed, so that in effect $(G,·)$ is determined by its right division operation. Here the analogous situation is examined for a Moufang loop. Subtleties arise which are not present in the group case since there is a choice of defining identities and the identities produced by...

We prove a version of Hrushovski's Socle Lemma for rigid groups in an arbitrary simple theory.