We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops. In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for asmany combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively.

Call a non-Moufang Bol loop minimally non-Moufang if every proper subloop is Moufang and minimally nonassociative if every proper subloop is associative. We prove that these concepts are the same for Bol loops which are nilpotent of class two and in which certain associators square to 1. In the process, we derive many commutator and associator identities which hold in such loops.

Let $L$ be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Let $f:L\to\{\pm1\}$ be a homomorphism and, for $\alpha=\sum\alpha_\ell\ell\in\mathbb{Z}L$, define $\alpha^f=\sum f(\ell)\alpha_\ell\ell^{-1}$. Call $\alpha$ f-unitary if $\alpha^f=\alpha^{-1}$ or $\alpha^f=-\alpha^{-1}$. In this paper, we identify the RA loops $L$ with the property that all units in $\mathbb{Z}L$ are $f$-unitary. Along the way, we extend a famous theorem of Higman to a case still undecided in group rings.

AMS 2000 Mathematics subject classification: Primary 20N05. Secondary 17D05; 16S34; 16U60