Groupoids arising from algebraic actions of semigroups are rigid: isomorphisms recover rational embeddings of the actions, and for rings of algebraic integers the groupoid determines the ring up to isomorphism.
Li, On K-theoretic invariants of semigroup C∗-algebras attached to number fields, Part II, Adv
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Algebraic actions II. Groupoid rigidity
Groupoids arising from algebraic actions of semigroups are rigid: isomorphisms recover rational embeddings of the actions, and for rings of algebraic integers the groupoid determines the ring up to isomorphism.