{"paper":{"title":"Basic and degenerate pregeometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cai Heng Li, Cheryl E. Praeger, Geoffrey Pearce, Michael Giudici","submitted_at":"2010-09-01T02:23:20Z","abstract_excerpt":"We study pairs $(\\Gamma,G)$, where $\\Gamma$ is a 'Buekenhout-Tits' pregeometry with all rank 2 truncations connected, and $G\\leqslant\\mathrm{Aut} \\Gamma$ is transitive on the set of elements of each type. \nThe family of such pairs is closed under forming quotients with respect to $G$-invariant type-refining partitions of the element set of $\\Gamma$. We identify the 'basic' pairs (those that admit no non-degenerate quotients), and show, by studying quotients and direct decompositions, that the study of basic pregeometries reduces to examining those where the group $G$ is faithful and primitive "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0075","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"}