  In this paper our aim is twofold. First, we introduce the notion of star
gluing of numerical semigroups and show that arithmetically Cohen-Macaulay and
Gorenstein properties of the projective closure are preserved under this gluing
operation. We then give a condition on Gr\"{o}bner basis of the defining ideal
of an affine monomial curve which ensures that the Betti sequence of the affine
curve is the same as the Betti sequence of its projective closure. We also
study the effect of simple gluing on Betti sequences of the projective closure.
Finally, we construct some numerical semigroups, using a gluing technique, such
that the Cohen-Macaulay type of corresponding affine curve and its projective
closure are both $n$.
