{"paper":{"title":"Two Generalizations of Homogeneity in Groups with Applications to Regular Semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Jo\\~ao Ara\\'ujo, Peter J. Cameron","submitted_at":"2012-04-10T15:39:14Z","abstract_excerpt":"Let $X$ be a finite set such that $|X|=n$ and let $i\\leq j \\leq n$. A group $G\\leq \\sym$ is said to be $(i,j)$-homogeneous if for every $I,J\\subseteq X$, such that $|I|=i$ and $|J|=j$, there exists $g\\in G$ such that $Ig\\subseteq J$. (Clearly $(i,i)$-homogeneity is $i$-homogeneity in the usual sense.)\n  A group $G\\leq \\sym$ is said to have the $k$-universal transversal property if given any set $I\\subseteq X$ (with $|I|=k$) and any partition $P$ of $X$ into $k$ blocks, there exists $g\\in G$ such that $Ig$ is a section for $P$. (That is, the orbit of each $k$-subset of $X$ contains a section fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.2195","kind":"arxiv","version":2},"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"}