Sub-semigroups determined by the zero-divisor graph

In this paper we study sub-semigroups of a finite or an infinite zero-divisor semigroup S   determined by properties of the zero-divisor graph Γ(S)Γ(S). We use these sub-semigroups to study the correspondence between zero-divisor semigroups and zero-divisor graphs. In particular, we discover a class of sub-semigroups of reduced semigroups and we study properties of sub-semigroups of finite or infinite semilattices with the least element. As an application, we provide a characterization of the graphs which are zero-divisor graphs of Boolean rings. We also study how local property of Γ(S)Γ(S) affects global property of the semigroup S, and we discover some interesting applications. In particular, we find that no finite or infinite two-star graph has a corresponding nil semigroup.