On the rank of monoids of endomorphisms of a finite directed path

In this paper, we consider endomorphisms of a finite directed path from monoid generators perspective. Our main aim is to determine the rank of the monoid [Formula: see text] of all weak endomorphisms of a directed path with [Formula: see text] vertices, which is a submonoid of the widely studied monoid [Formula: see text] of all order-preserving transformations of an [Formula: see text]-chain. Also, we describe the regular elements of [Formula: see text] and calculate its size and number of idempotents.