Alternative rings of small order and the hunt for Moufang circle loops

Let (R, +, ·) be a nilpotent alternative ring and, for x, y ∈ R, define x ○ y = x + y + xy. Then (R, o) is a Moufang loop called a “circle loop”. Circle groups, which are not hard to exhibit, have sparked the interest of a number of people for more than forty years. Relatively little attention has been directed at the not associative situation, however. Indeed, the scarcity of nilpotent alternative rings (which are not associative) makes even the discovery of Moufang circle loops difficult. The search for Moufang circle loops of small order has led to a recent enumeration of small order alternative rings and alternative algebras of small finite dimension which is described here.