Publication at Faculty of Mathematics and Physics |
2019
Abstract
Let Q be a quasigroup. Put a(Q) = vertical bar{(x, y, z) is an element of Q(3); x(yz)) = (xy)z}vertical bar and assume that vertical bar Q vertical bar = n.
Let delta(L) and delta(R) be the number of left and right translations of Q that are fixed point free. Put delta(Q) = delta(L) + delta(R).
Denote by i(Q) the number of idempotents of Q. It is shown that a(Q) >= 2n - i(Q) + delta(Q).
Call Q extremely nonassociative if a(Q) = 2n - i(Q). The paper reports what seems to be the first known example of such a quasigroup, with n = 8, a(Q) = 16, and i(Q) = 0.
It also provides supporting theory for a search that verified a(Q) >= 16 for all quasigroups of order 8.