The main concern of this paper are quasigroups of order nine that possess at most 18 associative triples. The order nine is the least order for which there exists a quasigroup Q ( , )
Let
Q be a quasigroup. Put
bolda
(
Q
)
=
∣
{
(
x
,
y
,
z
)
∈
Q
3
;
x
(
y
z
)
=
(
x
y
)
z
}
∣
and assume that
∣
Q
∣
=
n. Let
δ
L and
δ
R be the number of left and right translations of
Q that are fixed point free. Put
δ
(
Q
)
=
δ
normalL
+
δ
normalR. Denote by
i
(
Q
) the number of idempotents of
Q. It is shown that
bolda
(
Q
)
≥
2
n
−
i
(
Q
)
+
δ
(
Q
). Call
Q extremely nonassociative if
bolda
(
Q
)
=
2
n
−
i
(
Q
). The paper reports what seems to be the first known example of such a quasigroup, with
n
=
8,
bolda
(
Q
)
=
16, and
i
(
Q
)
=
0. It also provides supporting theory for a search that verified
bolda
(
Q
)
≥
16 for all quasigroups of order
8.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.