The annihilating-ideal graph of commutative semigroups

详细信息    查看全文

• 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 adc43b1046432c9d9a017c5f23e503b" 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 adc43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S)$\mathbb{AG}(S)$ if and only if 97f0f" title="Click to view the MathML source">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 971bbfd039685788f340bb546d3ce89" title="Click to view the MathML source">A^⁎(S)${\mathbb{A}}^{⁎}(S)$ are characterized. Connections are drawn between adc43b1046432c9d9a017c5f23e503b" 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 adc43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S)$\mathbb{AG}(S)$ are described. Semigroups S   for which adc43b1046432c9d9a017c5f23e503b" 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 ad5c05f34bee6fc" title="Click to view the MathML source">χ(AG(S))$\chi (\mathbb{AG}(S))$ are given for a general commutative semigroup.