Some familiar graphs on the rings of measurable functions

Abstract: 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 $\mathcal{M}(X, \mathcal{A})$.