AI 中文总结
研究非交换半群的有向扩展零因子图\(\overrightarrow{\Gamma}_E(S)\)，通过幂零元等代数概念，探讨其与零因子图\(\overrightarrow{\Gamma}(S)\)的关系，包括生成子图、相等条件、含圈及直径情况，还研究了特定顶点，得出相关结果。
AI 中文摘要
本文引入非交换半群\(S\)的有向扩展零因子图\(\overrightarrow{\Gamma}_E(S)\)，研究其与\(S\)的零因子图\(\overrightarrow{\Gamma}(S)\)的关系。证明了\(\overrightarrow{\Gamma}(S)\)是\(\overrightarrow{\Gamma}_E(S)\)的生成子图，得到\(\overrightarrow{\Gamma}_E(S)=\overrightarrow{\Gamma}(S)\)的充要条件，确定\(\overrightarrow{\Gamma}_E(S)\)含圈的条件及直径严格小于\(\overrightarrow{\Gamma}(S)\)的条件，还研究了与其他所有顶点相邻或从其他所有顶点相邻的顶点。通过幂零元、幂零指数和元素的零化子集等代数概念得出结果。
英文摘要
In this paper, the directed extended zero-divisor graph $\overrightarrowΓ_E(S)$ of a non-commutative semigroup $S$ is introduced and the relationships between $\overrightarrowΓ_E(S)$ and the zero-divisor graph $\overrightarrowΓ(S)$ of $S$ are examined. The fact that $\overrightarrowΓ(S)$ is a spanning subgraph of $\overrightarrowΓ_E(S)$ is proved. Necessary and sufficient conditions for the equality $\overrightarrowΓ_E(S)=\overrightarrowΓ(S)$ are obtained. Moreover, the conditions under which $\overrightarrowΓ_E(S)$ contains a cycle were established. In addition, conditions under which the diameter of $\overrightarrowΓ_E(S)$ is strictly smaller than that of $\overrightarrowΓ(S)$ 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.
Comments21 pages, 3 figures, This paper is derived from the first author's master's thesis supervised by the second author. The third author contributed to the development of further results and to the writing of the paper
