{"paper":{"title":"Algebra in superextensions of groups, I: zeros and commutativity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.GN","authors_text":"O.Nykyforchyn, T.Banakh, V.Gavrylkiv","submitted_at":"2008-02-13T15:52:13Z","abstract_excerpt":"Given a group $X$ we study the algebraic structure of its superextension\n  $\\lambda(X)$. This is a right-topological semigroup consisting of all maximal linked systems on $X$ endowed with the operation $$\\mathcal A\\circ\\mathcal B=\\{C\\subset X:\\{x\\in X:x^{-1}C\\in\\mathcal B\\}\\in\\mathcal A\\}$$ that extends the group operation of $X$. We characterize right zeros of $\\lambda(X)$ as invariant maximal linked systems on $X$ and prove that $\\lambda(X)$ has a right zero if and only if each element of $X$ has odd order. On the other hand, the semigroup $\\lambda(X)$ contains a left zero if and only if it "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0802.1853","kind":"arxiv","version":1},"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"}