{"paper":{"title":"Transformation Semigroups Which Are Disjoint Union of Symmetric Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Kritsada Sangkhanan, Utsithon Chaichompoo","submitted_at":"2024-11-22T17:17:54Z","abstract_excerpt":"Let $X$ be a nonempty set and $T(X)$ the full transformation semigroup on $X$. For any equivalence relation $E$ on $X$, define a subsemigroup $T_{E^*}(X)$ of $T(X)$ by\n  $$\n  T_{E^*}(X)=\\{\\alpha\\in T(X):\\text{for all}\\ x,y\\in X, (x,y)\\in E\\Leftrightarrow (x\\alpha,y\\alpha)\\in E\\}.\n  $$\n  We have the regular part of $T_{E^*}(X)$, denoted by $\\mathrm{Reg}(T)$, is the largest regular subsemigroup of $T_{E^*}(X)$. Defined the subsemigroup $Q_{E^*}(X)$ of $T_{E^*}(X)$ by\n  $$\n  Q_{E^*}(X)=\\{\\alpha\\in T_{E^*}(X):|A\\alpha|=1\\ \\text{and}\\ A\\cap X\\alpha\\neq\\emptyset\\ \\text{for all}\\ A\\in X/E\\}.\n  $$\n  T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.15081","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.15081/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}