{"paper":{"title":"On the Dominions of Certain Semigroups of Transformations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Halima H. Assiri, Jehan A. Albar (King Abdulaziz University)","submitted_at":"2026-06-26T16:51:35Z","abstract_excerpt":"In the full transformation semigroup $T_n$ on a finite chain $X_n$, let $D_n=\\{\\alpha \\in T_n:(\\forall x \\in X_n) \\ x\\alpha \\leq x\\}$ be the subsemigroup of all order-decreasing maps of $T_n$, and let $O_n=\\{\\alpha \\in T_n:(\\forall x ,y\\in X_n) \\ x \\leq y \\Rightarrow x\\alpha \\leq y\\alpha\\}$ be the subsemigroup of all order-preserving maps of $T_n$. The Catalan monoid $C_n$ is a semigroup of all order-decreasing and order-preserving full transformations of $X_n$. In this paper, it is shown that $O_n$ is closed in $T_n$. Also, the dominion of $D_n$ and the dominion of $C_n$ in $T_n$, denoted by "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28477","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/2606.28477/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"}