Set-theoretic complete intersection lattice ideals in monoid rings

Assume that the characteristic of the base field is zero. We give a necessary and sufficient condition for the defining ideal of a monomial curve to be generated by one polynomial on another lattice ideal, up to radical. By applying it, we provide two examples of monomial curves in affine 4-space; in the first one, a monomial curve is set-theoretic complete intersection, in the second, it is never generated by two binomials and one polynomial up to radical.