The monoid of all order-preserving k-extensive full transformations

Let 1 <= k <= n. We denote by O epsilon(n,k) the monoid of all order-preserving k-extensive full transformations on {1,& mldr;,n} ordered in the standard way. In this paper, it is shown that O epsilon(n,k) is not a regular semigroup in general, but it is an abundant semigroup. Moreover, we prove that O epsilon(n,k) is idempotent-generated and compute the idempotent rank and rank of the monoid O epsilon(n,k). Moreover, we determine the maximal idempotent generated subsemigroups of the monoid O epsilon(n,k).