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Moreover, we give presentations of [Formula: see text] on [Formula: see text] generators and [Formula: see text] relations and on [Formula: see text] generators and [Formula: see text] relations. We also consider the remarkable inverse submonoid [Formula: see text] of [Formula: see text] constituted by all its order-preserving transformations. We show that [Formula: see text] has rank [Formula: see text] and [Formula: see text] elements. Furthermore, we exhibit presentations of [Formula: see text] on [Formula: see text] generators and [Formula: see text] relations and on [Formula: see text] generators and [Formula: see text] relations. <\/jats:p>","DOI":"10.1142\/s1793557124500177","type":"journal-article","created":{"date-parts":[[2024,2,18]],"date-time":"2024-02-18T04:22:00Z","timestamp":1708230120000},"source":"Crossref","is-referenced-by-count":2,"title":["On the cyclic inverse monoid on a finite set"],"prefix":"10.1142","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1057-4975","authenticated-orcid":false,"given":"V\u00edtor H.","family":"Fernandes","sequence":"first","affiliation":[{"name":"Center for Mathematics and Applications (NOVA Math) and Department of Mathematics, Faculdade de Ci\u00eancias e Tecnologia, Universidade Nova de Lisboa, Monte da Caparica, 2829-516 Caparica, Portugal"}]}],"member":"219","published-online":{"date-parts":[[2024,3,6]]},"reference":[{"key":"S1793557124500177BIB001","first-page":"261","volume":"45","author":"A\u012dzen\u0161tat A. 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