Nilpotent Families of Endomorphisms of ( [formula omitted]( V) +,∪)

Abstract

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: • find bounds 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); • find a way to demonstrate non-nilpotency of such semigroups (find as simple as possible characterization of non-nice graphs) .