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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303210&_mathId=si1.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=2adc43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S)class="mathContainer hidden">class="mathCode">$\mathbb{AG}(S)$, the annihilating-ideal graph, to a commutative semigroup S  . This graph has vertex set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303210&_mathId=si2.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=099e30e9674453fa156f55fcebb1dd27" title="Click to view the MathML source">A^⁎(S)=A(S)∖{(0)}class="mathContainer hidden">class="mathCode">${\mathbb{A}}^{⁎}(S)=\mathbb{A}(S)\setminus \{(0)\}$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303210&_mathId=si3.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=f1695d3157ea4c03cbd3f01eb595b20c" title="Click to view the MathML source">A(S)class="mathContainer hidden">class="mathCode">$\mathbb{A}(S)$ is the set of proper ideals of S   with nonzero annihilator. Two distinct vertices class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303210&_mathId=si313.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=536ec557dcedcc40b911e4741b398190" title="Click to view the MathML source">I,J∈A^⁎(S)class="mathContainer hidden">class="mathCode">$I,J\in {\mathbb{A}}^{⁎}(S)$ are defined to be adjacent in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303210&_mathId=si1.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=2adc43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S)class="mathContainer hidden">class="mathCode">$\mathbb{AG}(S)$ if and only if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303210&_mathId=si5.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=0a2d4dfc0f93db2e8d7f321671f97f0f" title="Click to view the MathML source">IJ=(0)class="mathContainer hidden">class="mathCode">$IJ=(0)$, the zero ideal. Conditions are given to ensure a finite graph. Semigroups for which each nonzero, proper ideal of S   is an element of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303210&_mathId=si6.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=3971bbfd039685788f340bb546d3ce89" title="Click to view the MathML source">A^⁎(S)class="mathContainer hidden">class="mathCode">${\mathbb{A}}^{⁎}(S)$ are characterized. Connections are drawn between class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303210&_mathId=si1.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=2adc43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S)class="mathContainer hidden">class="mathCode">$\mathbb{AG}(S)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303210&_mathId=si117.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=033e8fc213e87201afc6f20cf84826d8" title="Click to view the MathML source">Γ(S)class="mathContainer hidden">class="mathCode">$\mathrm{\Gamma }(S)$, the well-known zero-divisor graph, and the connectivity, diameter, and girth of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303210&_mathId=si1.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=2adc43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S)class="mathContainer hidden">class="mathCode">$\mathbb{AG}(S)$ are described. Semigroups S   for which class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303210&_mathId=si1.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=2adc43b1046432c9d9a017c5f23e503b" title="Click to view the MathML source">AG(S)class="mathContainer hidden">class="mathCode">$\mathbb{AG}(S)$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303210&_mathId=si395.gif&_user=111111111&_pii=S0021869316303210&_rdoc=1&_issn=00218693&md5=7e15108d72d68750bad5c05f34bee6fc" title="Click to view the MathML source">χ(AG(S))class="mathContainer hidden">class="mathCode">$\chi (\mathbb{AG}(S))$ are given for a general commutative semigroup.