Artin's equivalence generalized (Q1001458)

It is known that non-zero fractional ideals of a Dedekind domain form a group under multiplication. There exist more general domains where the non-zero fractional ideals form a Clifford semigroup under multiplication. Artin introduced an equivalence relation on the set of non-zero fractional ideals such that in certain integral domains the equivalence classes formed a group under multiplication. In this work, the authors introduce a more general equivalence relation such that in certain integral domains the equivalence classes of non-zero fractional ideals form a Clifford semigroup. They adapted the same abstract ideal-theoretic approach of discussing Artin's relation in more general terms for lattice-ordered monoids with residuals and modified this to their generalized Artin relation in residual monoids (not necessarily lattice-ordered). In such a monoid \(S\), relatively divisorial elements are introduced, and conditions ensuring that \(S\) admits a homomorphism onto the semigroup of relatively divisorial elements (when this semigroup is a Clifford monoid) are obtained.