On the cyclic inverse monoid on a finite set

Abstract

In this paper, we study the cyclic inverse monoid [Formula: see text] on a set [Formula: see text] with [Formula: see text] elements, i.e. the inverse submonoid of the symmetric inverse monoid on [Formula: see text] consisting of all restrictions of the elements of a cyclic subgroup of order [Formula: see text] acting cyclically on [Formula: see text]. We show that [Formula: see text] has rank [Formula: see text] (for [Formula: see text]) and [Formula: see text] elements. 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.