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The zero-divisor graph of a commutative semigroup. (English) Zbl 1011.20056
Let $S$ be a commutative multiplicative semigroup with $0$ ($0x=0$ for all $x\in S$). In this paper, the authors introduce and investigate the zero-divisor graph of $S$, denoted by $\Gamma(S)$. In analogy with the recently studied zero-divisor graph of a commutative ring, the vertices of $\Gamma(S)$ are the nonzero zero-divisors of $S$, and two distinct vertices $x$ and $y$ are connected by an edge if $xy=0$. They give several results about the shape of $\Gamma(S)$. For example, $\Gamma(S)$ is always connected and the diameter of $\Gamma(S)\le 3$. The graphs without a cycle which can be realized by some $\Gamma(S)$ are determined. If $\Gamma(S)$ contains a cycle, then the core of $\Gamma(S)$ is a union of squares and triangles, and any vertex not in the core is an end which is connected to the core by a single edge.

20M14Commutative semigroups
05C25Graphs and abstract algebra
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