Clifford semigroups of ideals in monoids and domains

We investigate the ideal semigroup and the ideal class semigroup built by the fractional ideals of an ideal system on a monoid or on a domain. We provide criteria for these semigroups to be Clifford semigroups or Boolean semigroups. In particular, we consider the case of valuation monoids (domains) and of Prufer-like monoids (domains). By the way, we prove that a monoid (domain) is of Krull type if every locally principal ideal is finite.