{"paper":{"title":"Semigroup of transformations with restricted partial range: Regularity, abundance and some combinatorial results","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Jiulin Jin, Taijie You","submitted_at":"2023-06-01T15:43:48Z","abstract_excerpt":"Suppose that $X$ be a nonempty set. Denote by $\\mathcal{T}(X)$ the full transformation semigroup on $X$. For $\\varnothing \\neq Z\\subseteq Y\\subseteq X$, let $\\mathcal{T}(X,Y,Z)=\\{\\alpha \\in \\mathcal{T}(X): Y\\alpha \\subseteq Z \\}$. Then $\\mathcal{T}(X,Y,Z)$ is a subsemigroup of $\\mathcal{T}(X)$. In this paper, we characterize the regular elements of the semigroup $\\mathcal{T}(X,Y,Z)$, and present a necessary and sufficient condition under which $\\mathcal{T}(X,Y,Z)$ is regular. Furthermore, we investigate the abundance of the semigroup $\\mathcal{T}(X,Y,Z)$ for the case $Z\\subsetneq Y\\subsetneq X"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.00822","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/2306.00822/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"}