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Necessary and sufficient conditions for $T_{RE}(X,X')$ and $T_{RE}(Y,Y')$ to be isomorphic are given. This isomorphism theorem is applied to classify the semigroup of regressive transformations $T_{RE}(X,X')$ where $X$ are familiar subchains of $\\R$, the chain of real numbers."},"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":"1105.4922","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-05-25T02:28:39Z","cross_cats_sorted":[],"title_canon_sha256":"898a63b34d6b770b2ab6e4d4fc7ffb0449054c702c25d43b2351b349e1f27e52","abstract_canon_sha256":"7c2c26116ff7e6dd9e191d0c2c86111eec8fcaf5f1240652ffca5d50fed4cb51"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:04.711072Z","signature_b64":"S0Oz9XDxXPjWrO2oB7rYwUSLCCkz/iI2YgKaNM19osOcAsVqxZCynvJVXDK3bpCkd4V36yR0kQi1M5SCRyQdDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"410107466bbe6d29882571c17d73f00c76884dc6d784fdb7ce445f6630684adb","last_reissued_at":"2026-05-18T03:44:04.710154Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:04.710154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Classification of Regressive Transformation Semigroups on Chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Patanee Udomkavanich, Phichet Jitjankarn","submitted_at":"2011-05-25T02:28:39Z","abstract_excerpt":"For each subchain $X'$ of a chain $X$, let $T_{RE}(X, X')$ denote the semigroup under composition of all full regressive transformations, $\\alpha:X\\rightarrow X'$ satisfying $x\\alpha\\leq x$ for all $x\\in X$. 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