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In the current paper, we study module amenability, $n$-weak module amenability and module Arens regularity of the triangular Banach algebra $\\mathcal T=[\n{cc}\n  \\mathcal A & \\mathcal M\n  & \\mathcal B\n]$ (as an $\\mathfrak T:=\\Big{[\n{cc}\n  \\alpha &\n  & \\alpha\n] | \\alpha\\in\\mathfrak\nA\\Big}$-module). We employ these results to prove that for an inverse semigroup $S$ with subsemigroup $E$ of i"},"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":"1301.3237","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-01-15T06:50:14Z","cross_cats_sorted":[],"title_canon_sha256":"798855c28160eae4107e01fdc4c8281b1dff0c58164ab44b8cf7786aa1f4dcbf","abstract_canon_sha256":"9106f3e06ef18f79c1d0ec36140e172360f74043c0d034598c233e24eda5c690"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:36:26.132495Z","signature_b64":"9l+KFeEAot7MwiGfvFhRup/oS+JAydduQoreA/BO2O7k2D66tXUKDuJrlMWiIbeS1a1R2FpWl6MuVYoYSFvlCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe335132e175eb9865149010d150f936089b82a24599fbce3e57316ef29b2d37","last_reissued_at":"2026-05-18T03:36:26.131802Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:36:26.131802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"n-Weak Module Amenability of Triangular Banach Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Abasalt Bodaghi, Ali Jabbari","submitted_at":"2013-01-15T06:50:14Z","abstract_excerpt":"Let $\\mathcal A$, $\\mathcal B$ be Banach $\\mathfrak A$-modules with compatible actions and $\\mathcal M$ be a left Banach $\\mathcal A$-$\\mathfrak A$-module and a right Banach $\\mathcal B$-$\\mathfrak A$-module. 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