  In this paper, we study a family of binomial ideals defining monomial curves
in the $n-$dimensional affine space determined by $n$ hypersurfaces of the form
$x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}} \in k[x_1, \ldots, x_n]$ with
$u_{ii} = 0$, $i\in \{ 1, \ldots, n\}$. We prove that, the monomial curves in
that family are set-theoretic complete intersection. Moreover, if the monomial
curve is irreducible, we compute some invariants such as genus, type and
Fr\"obenius number of the corresponding numerical semigroup. We also describe a
method to produce set-theoretic complete intersection semigroup ideals of
arbitrary large height.
