%0 Journal Article %T Distributive lattices and some related topologies in comparison with zero-divisor graphs %J Categories and General Algebraic Structures with Applications %I Shahid Beheshti University %Z 2345-5853 %A Bagheri, Saeid %A Koohi Kerahroodi, mahtab %D 2021 %\ 01/01/2021 %V 14 %N 1 %P 223-244 %! Distributive lattices and some related topologies in comparison with zero-divisor graphs %K Distributive lattice %K Goldie dimension %K compressed zero-divisor graph %K domination number %R 10.29252/cgasa.14.1.223 %X 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 $\mathcal L$. %U https://cgasa.sbu.ac.ir/article_94188_8886cc33fcfa95bf3e8a98c714546517.pdf