Endomorphisms of functional graphs

A functional graph is a digraph describing the action of a function on a set. The endomorphisms of such graphs are of interest in group theory: the centraliser of an element of the semigroup of all mappings of a set into itself coincides with the set of all endomorphisms of a digraph corresponding to this element.
In this study, the functional graphs with a countable set of vertices are considered. We introduce the notion of lineals and calculate the cardinality of the set of endomorphisms of functional graphs which are not lineals.