Directed Extended Zero Divisor Graphs Of Non-Commutative Semigroups

In this paper, the directed extended zero-divisor graph $\overrightarrow{\Gamma}_E(S)$ of a non-commutative semigroup $S$ is introduced and the relationships between $\overrightarrow{\Gamma}_E(S)$ and the zero-divisor graph $\overrightarrow{\Gamma}(S)$ of $S$ are examined. The fact that $\overrightarrow{\Gamma}(S)$ is a spanning subgraph of $\overrightarrow{\Gamma}_E(S)$ is proved. Necessary and sufficient conditions for the equality $\overrightarrow{\Gamma}_E(S)=\overrightarrow{\Gamma}(S)$ are obtained. Moreover, the conditions under which $\overrightarrow{\Gamma}_E(S)$ contains a cycle were established. In addition, conditions under which the diameter of $\overrightarrow{\Gamma}_E(S)$ is strictly smaller than that of $\overrightarrow{\Gamma}(S)$ are characterized. The vertices that are adjacent to or adjacent from all other vertices are investigated. The results are established by using several algebraic notions such as nilpotent elements, their nilpotency indices, and annihilator sets of elements.