The annihilating-ideal graph of commutative semigroups

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• 作者：Lisa DeMeyer^a ; ^{demey1la@cmich.edu" class="auth_mail" title="E-mail the corresponding author} ; Anna Schneider^b ; ^{annaschneider927@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：20M14 ; 20M12 ; 05C75 ; 05C15
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：469
• 期：Complete
• 页码：402-420
• 全文大小：392 K

文摘

In this paper, we associate an undirected graph c43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S)$\mathbb{AG}(S)$, the annihilating-ideal graph, to a commutative semigroup S  . This graph has vertex set A^⁎(S)=A(S)∖{(0)}${\mathbb{A}}^{⁎}(S)=\mathbb{A}(S)\setminus \{(0)\}$, where A(S)$\mathbb{A}(S)$ is the set of proper ideals of S   with nonzero annihilator. Two distinct vertices I,J∈A^⁎(S)$I,J\in {\mathbb{A}}^{⁎}(S)$ are defined to be adjacent in c43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S)$\mathbb{AG}(S)$ if and only if IJ=(0)$IJ=(0)$, the zero ideal. Conditions are given to ensure a finite graph. Semigroups for which each nonzero, proper ideal of S   is an element of A^⁎(S)${\mathbb{A}}^{⁎}(S)$ are characterized. Connections are drawn between c43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S)$\mathbb{AG}(S)$ and Γ(S)$\mathrm{\Gamma }(S)$, the well-known zero-divisor graph, and the connectivity, diameter, and girth of c43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S)$\mathbb{AG}(S)$ are described. Semigroups S   for which c43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S)$\mathbb{AG}(S)$ is a complete or star graph are characterized. Finally, it is proven that the chromatic number is equal to the clique number of the annihilating ideal graph for each reduced semigroup and null semigroup. Upper and lower bounds for χ(AG(S))$\chi (\mathbb{AG}(S))$ are given for a general commutative semigroup.