We consider algebras over a field
K
K
presented by generators
x
1
,
…
,
x_1,\dots ,
x
n
x_n
and subject to
(
n
2
)
n\choose 2
square-free relations of the form
x
i
x
j
=
x
k
x
l
x_{i}x_{j}=x_{k}x_{l}
with every monomial
x
i
x
j
,
i
≠
j
x_{i}x_{j}, i\neq j
, appearing in one of the relations. It is shown that for
n
>
1
n>1
the Gelfand-Kirillov dimension of such an algebra is at least two if the algebra satisfies the so-called cyclic condition. It is known that this dimension is an integer not exceeding
n
n
. For
n
≥
4
n\geq 4
, we construct a family of examples of Gelfand-Kirillov dimension two. We prove that an algebra with the cyclic condition with generators
x
1
,
…
,
x
n
x_1,\dots ,x_n
has Gelfand-Kirillov dimension
n
n
if and only if it is of
I
I
-type, and this occurs if and only if the multiplicative submonoid generated by
x
1
,
…
,
x
n
x_1,\dots ,x_n
is cancellative.