{"paper":{"title":"Congruences on Direct Products of Transformation and Matrix Monoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Gracinda Gomes, Jo\\~ao Ara\\'ujo, Wolfram Bentz","submitted_at":"2016-02-19T22:46:02Z","abstract_excerpt":"Malcev described the congruences of the monoid $T_n$ of all full transformations on a finite set $X_n=\\{1, \\dots,n\\}$. Since then, congruences have been characterized in various other monoids of (partial) transformations on $X_n$, such as the symmetric inverse monoid $In_n$ of all injective partial transformations, or the monoid $PT_n$ of all partial transformations.\n  The first aim of this paper is to describe the congruences of the direct products $Q_m\\times P_n$, where $Q$ and $P$ belong to $\\{T, PT,In\\}$.\n  Malcev also provided a similar description of the congruences on the multiplicative"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.06339","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"}