Abstract:
Quasigroups satisfying the identityx(x·
xy
) =
y
are called
π
-quasigroups
of type
T
1
. The spectrum of the defining identity is precisely
q
= 0 or 1(mod 3),
except for
q
= 6. Necessary conditions when a finite
π
-quasigroup of type
T
1
has the
order
q
= 0 (mod 3), are given. In particular, it is proved that a finite
π
-quasigroup
of type
T
1
such that the order of its inner mapping group is not divisible by three
has a left unit. Necessary and sufficient conditions when the identityx
(
x
·
xy
) =
y
is
invariant under the isotopy of quasigroups (loops) are found
