The monoid of all order-preserving <i>k</i>-extensive full transformations

Let [Formula: see text]. We denote by [Formula: see text] the monoid of all order-preserving k-extensive full transformations on [Formula: see text] ordered in the standard way. In this paper, it is shown that [Formula: see text] is not a regular semigroup in general, but it is an abundant semigroup. Moreover, we prove that [Formula: see text] is idempotent-generated and compute the idempotent rank and rank of the monoid [Formula: see text]. Moreover, we determine the maximal idempotent generated subsemigroups of the monoid [Formula: see text].