A description of a class of finite semigroups that are near to being Malcev nilpotent

In this paper we continue the investigations on the algebraic structure of a finite semigroup $S$ that is determined by its associated upper non-nilpotent graph $\mathcal{N}_{S}$. The vertices of this graph are the elements of $S$ and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). We introduce a class of semigroups in which the Mal'cev nilpotent property lifts through ideal chains. We call this the class of \B\ semigroups. The definition is such that the global information that a semigroup is not nilpotent induces local information, i.e. some two-generated subsemigroups are not nilpotent. It turns out that a finite monoid (in particular, a finite group) is \B\ if and only if it is nilpotent. Our main result is a description of \B\ finite semigroups $S$ in terms of their associated graph ${\mathcal N}_{S}$. In particular, $S$ has a largest nilpotent ideal, say $K$, and $S/K$ is a 0-disjoint union of its connected components (adjoined with a zero) with at least two elements.