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Zambella we consider $e(\\mathbb{D})=\\{\\mathbb{D}^\\prime\\mid (\\mathbb{M},\\mathbb{D})\\equiv (\\mathbb{M},\\mathbb{D}^\\prime)\\}$ and $o(\\mathbb{D})=\\{\\mathbb{D}^\\prime\\mid (\\mathbb{M},\\mathbb{D})\\cong (\\mathbb{M},\\mathbb{D}^\\prime)\\}$. The general question we ask is when $e(\\mathbb{D})=o(\\mathbb{D})$ ? The case where $\\mathbb{D}$ is $A$-invariant for some small set $A$ is rather straightforward: it just mean that $\\mathbb{D}$ is definable. We investigate the case whe"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1702.01892","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-02-07T06:35:59Z","cross_cats_sorted":[],"title_canon_sha256":"3a5bfb34ff24040344ad27cea6bd1a4b5fb968a5af9a5f48f36b2f6fea9d362e","abstract_canon_sha256":"995f8a17cae3ba25c8328e883ed6e0b8b82fba5e204c0986ca3e5042e95b305e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:20.784120Z","signature_b64":"5EKz97RVAP4DZjsq3ED7jDj0eTtFtHA1Zt9DEuVrbW6KnTzzOsJe04OxrRKJQzbdDpNA0xen9tHkZ1n4ploJCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dba73e4802a38febb2a6cbfcf12927e8b42cf8f5415d21a78616d3fb8fd66df3","last_reissued_at":"2026-05-18T00:41:20.783395Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:20.783395Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Orbits of subsets of the monster model and geometric theories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Enrique Casanovas, Luis Jaime Corredor","submitted_at":"2017-02-07T06:35:59Z","abstract_excerpt":"Let $\\mathbb{M}$ be the monster model of a complete first-order theory $T$. If $\\mathbb{D}$ is a subset of $\\mathbb{M}$, following D. Zambella we consider $e(\\mathbb{D})=\\{\\mathbb{D}^\\prime\\mid (\\mathbb{M},\\mathbb{D})\\equiv (\\mathbb{M},\\mathbb{D}^\\prime)\\}$ and $o(\\mathbb{D})=\\{\\mathbb{D}^\\prime\\mid (\\mathbb{M},\\mathbb{D})\\cong (\\mathbb{M},\\mathbb{D}^\\prime)\\}$. The general question we ask is when $e(\\mathbb{D})=o(\\mathbb{D})$ ? The case where $\\mathbb{D}$ is $A$-invariant for some small set $A$ is rather straightforward: it just mean that $\\mathbb{D}$ is definable. 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