Wilf inequality is preserved under gluing of semigroups

Wilf Conjecture on numerical semigroups is a question posed by Wilf in 1978 and 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 this Wilf inequality is preserved under gluing of numerical semigroups. If the numerical semigroups minimally generated by A = {a(1),& mldr;,a(p)} and B = {b(1),& mldr;,b(q)} satisfy the Wilf inequality, then so does their gluing which is minimally generated by C = k(1)A & sqcup; k(2)B. We discuss the extended Wilf's Conjecture in higher dimensions for certain affine semigroups and prove an analogous result.