Ideal-based zero-divisor graph of MV-algebras

TL;DR

This paper introduces an ideal-based zero-divisor graph for MV-algebras, analyzes its properties such as connectivity and diameter, and uses these graphs to classify MV-algebras into distinct types.

Contribution

It defines a new graph structure for MV-algebras based on ideals, explores its properties, and applies it to classify MV-algebras.

Findings

The graph is always connected with diameter ≤ 3.

Relationships between the graph's diameter and girth are established.

MV-algebras are classified into types based on these graph properties.

Abstract

Let $(A, \oplus, *, 0)$ be an MV-algebra, $(A, \odot, 0)$ be the associated commutative semigroup, and $I$ be an ideal of $A$. Define the ideal-based zero-divisor graph $\Gamma_{I}(A)$ of $A$ with respect to $I$ to be a simple graph with the set of vertices $V(\Gamma_{I}(A))=\{x\in A\backslash I ~|~ (\exists~ y\in A\backslash I) ~x\odot y\in I\},$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if $x\odot y\in I$. We prove that $\Gamma_{I}(A)$ is connected and its diameter is less than or equal to $3$. Also, some relationship between the diameter (the girth) of $\Gamma_{I}(A)$ and the diameter (the girth) of the zero-divisor graph of $A/I$ are investigated. And using the girth of zero-divisor graphs (resp. ideal-based zero-divisor graphs) of MV-algebras, we classify all MV-algebras into $2~($resp. $3)$ types.