The zero-divisor graphs of MV-algebras

Abstract

In this paper, we will introduce and study the zero-divisor graphs of MV-algebras. Let $(A, \oplus , *, 0)$ be an MV-algebra, and $(A, \odot , 0)$ be the associated semigroup. Define the zero-divisor graph $\Gamma (A)$ of A to be the simple graph with vertices $V(\Gamma (A))=\{x\in A ~|~ (\exists ~y\in A {\setminus } \{0\}) ~x\odot y=0\}$, and edges $E(\Gamma (A))=\{\text {the edge with ends } x~ \text {and } y ~|~ (x\ne y, x,y\in A) ~x\odot y=0\}$. We show that $\Gamma (A)$ is connected with $diam(\Gamma (A))\le 3$, where $diam(\Gamma (A))$ denotes the diameter of $\Gamma (A)$. Moreover, we characterize A with $diam(\Gamma (A))$ equal to 0, 1, 2 or 3. Finally, using the zero-divisor graph, we classify all MV-algebras of cardinality up to seven.