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Recently, it was shown in \\cite{I} that they admit left-right symmetric characterizations analogue to those of Frobenius algebras: a coalgebra $C$ is co-Frobenius if and only if it is isomorphic to its rational dual. We consider the more general quasi-co-Frobenius (QcF) coalgebras; in the first main result we show that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or equivalently right) rational dual $Rat(C^*)$, in the sense that certain coproduct or prod"},"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":"0803.0775","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2008-03-06T02:02:02Z","cross_cats_sorted":["math.CT","math.KT","math.RA","math.RT"],"title_canon_sha256":"fdef3d85cf66a126d68f36797b129ad7aa9fd757c7c9dd020a799d2d500cc74d","abstract_canon_sha256":"86f046fc562dc2eea79fe74a814fbb815e338d26ceaa57803953108c123001a3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:06.077597Z","signature_b64":"wPbcOXIvsVqDoelF8k4T+DNzkQz7jvCYz66J63O24PBMtbKgmz3vAngGHd5fgPFXInfb2ffk7IUO7U79nf15AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9a722dc60bc72627d90b4ebe196a00e966022b21f1abfc97c51773538cf2bf4","last_reissued_at":"2026-05-18T04:41:06.076900Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:06.076900Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generalized Frobenius Algebras and the Theory of Hopf Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.KT","math.RA","math.RT"],"primary_cat":"math.QA","authors_text":"Miodrag C. 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