Ideal-based zero-divisor graph of MV-algebras

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