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.
Neukirch, Kennzeichnung der p-adischen und der endlichen algebraischen Zahlk¨ orper, (German) Invent
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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.