{"paper":{"title":"A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.RA"],"primary_cat":"math.AC","authors_text":"Alfred Geroldinger, Daniel Smertnig, David J. Grynkiewicz, Nicholas R. Baeth","submitted_at":"2014-04-29T07:44:04Z","abstract_excerpt":"Let $R$ be a ring and let $\\mathcal C$ be a small class of right $R$-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let $\\mathcal V (\\mathcal C)$ denote a set of representatives of isomorphism classes in $\\mathcal C$ and, for any module $M$ in $\\mathcal C$, let $[M]$ denote the unique element in $\\mathcal V (\\mathcal C)$ isomorphic to $M$. Then $\\mathcal V (\\mathcal C)$ is a reduced commutative semigroup with operation defined by $[M] + [N] = [M \\oplus N]$, and this semigroup carries all information about direct-sum decompositions of modules in $\\mathcal C"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.7264","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}