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Although not a unit in $_{A}\\mathbb{S}%_{A},$ the base semialgebra $A$ has properties of a semiunit (in a sense which we clarify in this note). Motivated by this interesting example, we investigate semiunital semimonoidal categories $(\\mathcal{V}%, \\bullet, I)$ as a framework for studying notions like semimonoids (semicomonoids) as well as a notion of monads (comonads) which we call $\\mathbb{J}$-monads ($\\mathbb{J}$-% como"},"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":"1209.4114","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2012-09-18T22:29:14Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"acd7a86d796bf60311f49893057c28454d468e12cdb9c375871931373cff4a09","abstract_canon_sha256":"702b1fe1d47c824617d09c88fc8af5ad7ae4fdbe4da421284c765a40c797aeb6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:35:38.224751Z","signature_b64":"KLxCTLQQnXtkTqmjL2kV6lxmsbf0iA4JIBv3Qk9oLL/fK655o8SH2hp9HTcOdzZ0yCVD7MTZIoPXQrHOWPcZDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"00b876d25f6d0f0ec1707a4ad211167486ab5087df39ebafcfc5f6e2ddd0d956","last_reissued_at":"2026-05-18T03:35:38.223849Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:35:38.223849Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Semiunital Semimonoidal Categories (Applications to Semirings and Semicorings)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.CT","authors_text":"Jawad Abuhlail","submitted_at":"2012-09-18T22:29:14Z","abstract_excerpt":"The category $_{A}\\mathbb{S}_{A}$ of bisemimodules over a semialgebra $A,$ with the so called Takahashi's tensor product $-\\boxtimes_{A}-,$ is semimonoidal but not monoidal. 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