Suppose we have a set X with a 1) binary operation #, 2) an identity e, and 3) a unique two-sided inverse for each element. Let's call (X,#) an I-magma. (There are all kinds of names for structures similar to this, like quasigroups and loops, but I haven't yet found an official name for this exact concept.) If an I-magma were also associative, it would be a group. But suppose that instead of being associative we ask that an I-magma also be alternative: ----- For all x, y in X: x(xy) = (xx)y, (xy)y = x(yy), x(yx) = (xy)x ----- One class of alternative I-magmas is Moufang loops (see < http://www.encyclopediaofmath.org/index.php/Moufang_loop >). But I haven't seen anything implying that alternative I-magmas must be Moufang loops, so I suspect this is not true. QUESTION: What are the smallest alternative I-magmas??? A paper states: "There are 13 [Moufang loops of size <= 31]: one of order 12, five of order 16, one of order 20, five of order 24, and one of order 28. (See < http://www.ams.org/journals/tran/1974-188-00/S0002-9947-1974-0330336-3/S0002... >.) So each of these must be an alternative I-magma. But are there others <= size 31 as well? --Dan