Fernandes, Vítor H. "
On the cyclic inverse monoid on a finite set."
Asian-European Journal of Mathematics (DOI 10.1142/S1793557124500177; Online 6 Mar 2024). 17.2 (2024): 2450017 (16 pages).
AbstractIn 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⩾2) and n⋅2^n−n+1 elements. Moreover, we give presentations of CI_n on n+1 generators and (n^2+3n+4)/2 relations and on 2 generators and (n^2−n+6)/2 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⋅2^n−2n−1 elements. Furthermore, we exhibit presentations of OCI_n on n+2 generators and (n^2+3n+8)/2 relations and on n generators and (n^2+3n)/2 relations.
Fernandes, Vítor H. "
On the monoid of partial isometries of a wheel graph."
Asian-European Journal of Mathematics (DOI 10.1142/S1793557123502388; Online 16 Dec 2023). 17.1 (2024): 2350238 (18 pages).
AbstractIn this paper, we consider the monoid DPW_n of all partial isometries of a wheel graph W_n with n+1 vertices. Our main objective is to determine the rank of DPW_n. In the process, we also compute the ranks of three notable subsemigroups of DPW_n. We also describe Green's relations of DPW_n and of its three considered subsemigroups.