Wilf Inequality is preserved under Gluing of Semigroups

Wilf Conjecture on numerical semigroups is an inequality connecting the Frobenius number, embedding dimension and the genus of the semigroup. The conjecture is still open in general. We prove that the Wilf inequality is preserved under gluing of numerical semigroups. If the numerical semigroups minimally generated by $A = \{ a_1, \ldots, a_p\}$ and $B = \{ b_1, \ldots, b_q\}$ satisfy the Wilf inequality, then so does their gluing which is minimally generated by $C =k_1A\sqcup k_2B$. We discuss the extended Wilf's Conjecture in higher dimensions and prove an analogous result.