Note on star-operations and semistar-operations

A subsemigroup ⊃≠{0} of a torsion-free abelian (additive) group is called a grading monoid. First we show that [He] holds for a grading monoid S. Especially every ideal of an integrally closed grading monoid S is divisorial if and only if S is a valuation semigroup with the principal maximal ideal. Next we discuss whether [MSu, Proposition 8] holds for n=4, namely, an integrally closed domain D of dimension 4 is or is not a valuation ring if and only if 5〓|Σ'(D)|〓9, where Σ'(D) is the set of semistar-operations on D. Next. we study semigroup version of [AA2], [Q1] and [Q2]. Next we study the question of [A]. Finally we show that every cancellation ideal of a grading monoid is