{"paper":{"title":"Subgroups of direct products closely approximated by direct sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.GR","math.IT"],"primary_cat":"math.GN","authors_text":"Dmitri Shakhmatov, Maria V. Ferrer, Salvador Hernandez","submitted_at":"2013-06-17T19:10:43Z","abstract_excerpt":"Let $I$ be an infinite set, $\\{G_i:i\\in I\\}$ be a family of (topological) groups and $G=\\prod_{i\\in I} G_i$ be its direct product. For $J\\subseteq I$, $p_{J}: G\\to \\prod_{j\\in J} G_j$ denotes the projection. We say that a subgroup $H$ of $G$ is: (i) \\emph{uniformly controllable} in $G$ provided that for every finite set $J\\subseteq I$ there exists a finite set $K\\subseteq I$ such that $p_{J}(H)=p_{J}(H\\cap\\bigoplus_{i\\in K} G_i)$; (ii) \\emph{controllable} in $G$ provided that $p_{J}(H)=p_{J}(H\\cap\\bigoplus_{i\\in I} G_i)$ for every finite set $J\\subseteq I$; (iii) \\emph{weakly controllable} in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.3954","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"}