A combinatorial problem and numerical semigroups

Let a  = ( a 1 , …,  a n ) and b  = ( b 1 , …,  b n ) be two n -tuples of positive integers, let X be a set of positive integers, and let g be a positive integer. In this work we show an algorithmic process in order to compute all the sets C of positive integers that fulfill the following conditions: The cardinality of C is equal to g ; If x ,  y  ∈ ℕ \ {0} and x  +  y  ∈  C , then C  ∩ { x ,  y } ≠ ∅ ; If x  ∈  C and ( x  −  b i ) / a i  ∈ ℕ \ {0} for some i  ∈ {1, …,  n } , then ( x  −  b i ) / a i  ∈  C ; X  ∩  C  = ∅ .