{"paper":{"title":"The regular part of transformation semigroups that preserve double direction equivalence relation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Kritsada Sangkhanan","submitted_at":"2023-06-15T08:14:27Z","abstract_excerpt":"Let $T(X)$ be the full transformation semigroup on a set $X$ under the composition of functions. For any equivalence relation $E$ on $X$, define a subsemigroup $T_{E^*}(X)$ of $T(X)$ by $$T_{E^*}(X)=\\{\\alpha\\in T(X):\\text{for all}\\ x,y\\in X, (x,y)\\in E\\Leftrightarrow (x\\alpha,y\\alpha)\\in E\\}.$$ In this paper, we show that the regular part of $T_{E^*}(X)$, denoted $\\mathrm{Reg}(T)$, is the largest regular subsemigroup of $T_{E^*}(X)$. Then its Green's relations and ideals are described. Moreover, we find the kernel of $\\mathrm{Reg}(T)$ which is a right group and can be written as a union of sym"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.08932","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/2306.08932/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"}