Some familiar graphs on the rings of measurable functions

TL;DR

This paper explores the graph structures of rings of measurable functions, revealing isomorphisms and structural properties of various graphs under different measure spaces, especially Lebesgue measure.

Contribution

It introduces modified graph definitions based on almost everywhere equality and investigates their structural relationships, including isomorphisms and conditions for equality.

Findings

Subgraphs of comaximal and zero-divisor graphs are always isomorphic.

In Lebesgue measure spaces, the comaximal and zero-divisor graphs are isomorphic.

Non-atomic measure spaces are characterized via graph properties.

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