Some familiar graphs on the rings of measurable functions

• Published in: arXiv.org
• Main Authors: Nandi, Pratip, Atasi Deb Ray, Acharyya, Sudip Kumar
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 04.07.2023
• Subjects: Graph theory; Graphs; Rings (mathematics)
• ISSN: 2331-8422

In this paper, replacing `equality' by 'equality almost everywhere' we modify several terms associated with the ring of measurable functions defined on a measure space \((X, \mathcal{A}, \mu)\) and thereby study the graph theoretic features of the modified comaximal graph, annihilator graph and the weakly zero-divisor graph of the said ring. The study reveals a structural analogy between the modified versions of the comaximal and the zero-divisor graphs, which prompted us to investigate whether these two graphs are isomorphic. Introducing a quotient-like concept, we find certain subgraphs of the comaximal graph and the zero-divisor graph of \(\mathcal{M}(X, \mathcal{A})\) and show that these two subgraphs are always isomorphic. Choosing \(\mu\) as a counting measure, we prove that even if these two induced graphs are isomorphic, the parent graphs may not be so. However, in case of Lebesgue measure space on \(\mathbb{R}\), we establish that the comaximal and the zero-divisor graphs are isomorphic. Observing that both of the comaximal and the zero-divisor graphs of the ring \(\mathcal{M}(X, \mathcal{A})\) are subgraphs of the annihilator graph of the said ring, we find equivalent conditions for their equalities in terms of the partitioning of \(X\) into two atoms. Moreover, the non-atomicity of the underlying measure space \(X\) is characterized through graph theoretic phenomena of the comaximal and the annihilator graph of \(\mathcal{M}(X, \mathcal{A})\).