Special Pairs in the Generating Subset of the Weierstrass Semigroup at a Pair

We discuss the structure of the Weierstrass semigroup at a pair of points on an algebraic curve. It is known that the Weierstrass semigroup at a pair (P, Q) contains the unique generating subset Γ(P, Q). We find some characterizations of the elements of Γ(P, Q) and prove that, for any point P on a curve, Γ(P, Q) consists of only maximal elements for all except for finitely many points Q ≠ P on the given curve. Also we obtain more results concerning special and nonspecial pairs.