On the cyclic inverse monoid on a finite set

2022/11/03 by Vítor H. Fernandes, Fernandes, Vitor Hugo
Computer Science · Mathematics · #20M05 #20M20 #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2211.02155

openalex publication_date 2022/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study the cyclic inverse monoid \CI_n on a set Ω_n with n elements, i.e. the inverse submonoid of the symmetric inverse monoid on Ω_n consisting of all restrictions of the elements of a cyclic subgroup of order n acting cyclically on Ω_n. We show that \CI_n has rank 2 (for n\geqslant2) and n2ⁿ-n+1 elements. Moreover, we give presentations of \CI_n on n+1 generators and (1)/(2)(n²+3n+4) relations and on 2 generators and (1)/(2)(n²-n+6) relations. We also consider the remarkable inverse submonoid \OCI_n of \CI_n constituted by all its order-preserving transformations. We show that \OCI_n has rank n and 3⋅ 2ⁿ-2n-1 elements. Furthermore, we exhibit presentations of \OCI_n on n+2 generators and (1)/(2)(n²+3n+8) relations and on n generators and (1)/(2)(n²+3n) relations.

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