Let
S
S
be a semigroup. Then by a theorem of Tully [7]:
S
S
is a commutative semigroup iff
a
b
=
b
n
a
m
ab = {b^n}{a^m}
for all
a
,
b
∈
S
a,b \in S
(
m
,
n
⩾
1
m,n \geqslant 1
, fixed). We prove the following:
S
S
is a commutative semigroup iff
a
b
=
b
n
(
a
,
b
)
a
m
(
a
,
b
)
ab = {b^{n(a,b)}}{a^{m(a,b)}}
for all
a
,
b
∈
S
a,b \in S
, where one of the exponents
n
(
a
,
b
)
n(a,b)
and
m
(
a
,
b
)
m(a,b)
is constant and the other is independent of
a
a
or
b
b
.