The monoid of all orientation-preserving and extensive partial transformations on a finite chain

Let POPEn be the monoid of all orientation-preserving and extensive partial trans-formations on n = {1, . . . , n}. In this paper, we characterize the structure of the generating sets of POPEn, and prove that each generating set of POPEn contains a minimal idempotent generating set of POPEn. Moreover, the minimal generating sets and minimal idempotent generating sets of POPEn coincide. As applications, we compute the number of distinct minimal (idempotent) generating sets of POPEn, and prove that both the rank and the idempotent rank of the monoid POPEn are equal to n2+n+2/2 . Finally, we determine the maximal subsemigroups as well as the maximal idempotent generated subsemigroups of the monoid POPEn.