Nilpotent Families of Endomorphisms of (P(V)+,∪)

A directed graph G=(V,A) is k-nice if for every u,v∈V (allowing t=v), and for every orientation of the edges of an undirected path of length k, there exists a u−v walk of length k in G whose orientation coincides with that of the given path. A graph is nice if it is k-nice for some k. We generalize this notion using the notion of a nilpotent semigroup of endomorphisms of (P(V)+,∪), and consider two basic problems:• ounds for the nilpotency class of such semigroups in terms of their generators (in the language of graphs: provided that a graph on vertices is nice, find the smallest such that is -nice); x02022; way to demonstrate non-nilpotency of such semigroups (find as simple as possible characterization of non-nice graphs)