The ideal and divisor class group of relevant arithmetic objects have been a central topic of investigations in algebra and number theory. On the contrary, the corresponding semigroups have only recently been studied more extensively. Bazzoni (2000) characterized integrally closed domains whose ideal class semigroup is a Clifford semigroup. In the case of Prüfer v-multiplication domains and general Prüfer monoids her results were generalized and strenghtened by Kabbaj-Mimouni (2007) and Halter-Koch (2009). Already in 1961, Dade, Taussky and Zassenhaus (DTZ) investigated the ideal class semigroup of orders over Dedekind domains and proved that they are pi-stable. Halter-Koch (2007) extended their results to the case of (class-)semigroup of lattices over Dedekind domains (in a fixed finite algebraic extension). In the present work, I generalize and replenish the results of DTZ and Halter-Koch. In the first part I study the structure of the r-ideal(class-)semigroups built by an arbitrary ideal system r on a monoid. In the second part I investigate the semigroup of semidivisorial lattices over a domain in a finite algebraic extension of its field of quotients (for Krull domains semidivisoriality coincides with reflexivity). Under suitable conditions on the underlying domains (which are in particular satisfied by Krull domains), I determine the idempotents and provide criteria for the pi*-stability of the domains and for the completeness of the investigated semigroups. The abstract theory is complemented by a series of examples, which in particular illustrate the reasonableness of the assumptions made in the theory. Among others, I present examples of two-dimensional noetherian domains having a v-idempotent conductor, and non-trivial examples of domains with an almost complete v-ideal semigroup.