The Number of Star Operations on Numerical Semigroups and on Related Integral Domains

We study the cardinality of the set Star(S) of star operations on a numerical semigroup S; in particular, we study ways to estimate Star(S) and to bound the number of nonsymmetric numerical semigroups such that |Star(S)|≤ n. We also study this problem in the setting of analytically irreducible, residually rational rings whose integral closure is a fixed discrete valuation ring