On the cyclic inverse monoid on a finite set
type: Item
instance of: scholarly article
title: On the cyclic inverse monoid on a finite set
author: Vítor H. Fernandes
published in: Asian-European Journal of Mathematics
publication date: 2024-12-09T00:00:00Z
review text: Let \(\Omega_n\) be a set with \(n\) elements and let \(\mathcal{CI}_n\) be the inverse submonoid of the symmetric inverse monoid on the set \(\Omega_n\) consisting of all restrictions of the elements of a cyclic subgroup of order \(n\) acting cyclically on \(\Omega_n\).\N\NIn this paper under review the author studies the monoid. He proves that the monoid \(\mathcal{CI}_n\) has rank \(2\) (for \(n\geqslant 2\)) and it has \(n2^n-n+1\) elements. The author gives presentations of \(\mathcal{CI}_n\) on \(n+1\) generators and \(\frac{1}{2}(n^2+3n+4)\) relations, and on \(2\) generators and \(\frac{1}{2}(n^2-n+6)\) relations. The author studies the remarkable inverse submonoid \({\mathcal{OCI}}_n\) of \(\mathcal{CI}_n\) constituted by all its order-preserving transformations. He shows that the monoid \({\mathcal{OCI}}_n\) has rank \(n\) and it contains \(3\cdot 2^n-2n-1\) elements. Furthermore, the author exhibits the presentations of the monoid \({\mathcal{OCI}}_n\) on \(n+2\) generators and \(\frac{1}{2} (n^2 + 3n + 8)\) relations, and on \(n\) generators and \(\frac{1}{2}(n^2 + 3n)\) relations.
reviewed by: O. Gutik
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 7956105
zbMATH Keywords: partial permutations
zbMATH Keywords: cyclic group
zbMATH Keywords: order-preserving
zbMATH Keywords: orientation-preserving
zbMATH Keywords: rank
zbMATH Keywords: presentations
MaRDI profile type: Publication
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DOI: 10.1142/S1793557124500177
description: scientific article; zbMATH DE number 7956105