{"paper":{"title":"On the semigroup of partial isometries of a finite chain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Abdullahi Umar, Rotimi Kehinde","submitted_at":"2010-12-30T07:51:14Z","abstract_excerpt":"Let ${\\cal I}_n$ be the symmetric inverse semigroup on $X_n = \\{1, 2,..., n\\}$ and let ${\\cal DP}_n$ and ${\\cal ODP}_n$ be its subsemigroups of partial isometries and of order-preserving partial isometries of $X_n$, respectively. In this paper we investigate the cycle structure of a partial isometry and characterize the Green's relations on ${\\cal DP}_n$ and ${\\cal ODP}_n$. We show that ${\\cal ODP}_n$ is a $0-E-unitary$ inverse semigroup. We also investigate the cardinalities of some equivalences on ${\\cal DP}_n$ and ${\\cal ODP}_n$ which lead naturally to obtaining the order of the semigroups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.0049","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"}