Distributive lattices and some related topologies in comparison with zero-divisor graphs

Document Type : Research Paper


Department of Mathematics, Malayer University, P.O.Box: 65719-95863, Malayer, Iran.


In this paper,
for a distributive lattice $\mathcal L$, we study and compare some lattice theoretic features of $\mathcal L$ and topological properties of the Stone spaces ${\rm Spec}(\mathcal L)$ and ${\rm Max}(\mathcal L)$ with the corresponding graph theoretical aspects of the zero-divisor graph $\Gamma(\mathcal L)$.
Among other things,
we show that the Goldie dimension of $\mathcal L$ is equal to the cellularity of the topological space ${\rm Spec}(\mathcal L)$ which is also equal to the clique number of the zero-divisor graph $\Gamma(\mathcal L)$. Moreover, the domination number of $\Gamma(\mathcal L)$ will be compared with the density and the weight of the topological space ${\rm Spec}(\mathcal L)$.

 For a $0$-distributive lattice $\mathcal L$, we investigate the compressed subgraph $\Gamma_E(\mathcal L)$ of the zero-divisor graph $\Gamma(\mathcal L)$ and determine some properties of this subgraph in terms of some lattice theoretic objects such as associated prime ideals of $\mathcal L$.


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