Star-operations on semigroups

Let S be a grading monoid with quotient group q(S) , let F(S) be the set of fractional ideals of S . For A ∈ F(S) , define Aw= {x ∈ q(S) \mid J+x \subseteq A for some f.g. ideal J of S with J-1=S} and A_ \overline w ={x ∈ q(S)\mid J+x \subseteq A for some ideal J of S with J-1=S} . Then w and \overline w are star-operations on F(S) such that w ≤ \overline w . Using these star-operations, we give characterizations of Krull semigroups and pre-Krull semigroups. Also we show that for every maximal * -ideal P of S , if SP is a valuation semigroup, then * -cancellation ideals are * -locally principal ideals, where * is a star-operation on S of finite character. Finally, we show that S is a pre-Krull semigroup (H-semigroup) if and only if the polynomial semigroup S[x] is a pre-Krull semigroup (H-semigroup).