Distributive lattices and some related topologies in comparison with zero-divisor graphs - Saeid Bagheri; mahtab Koohi Kerahroodi (Categories and General Algebraic Structures with Applications, 2021)
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 $Γ(\mathcal L)$.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 $Γ(\mathcal L)$. Moreover, the domination number of $Γ(\mathcal L)$ 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 $Γ_E(\mathcal L)$ of the zero-divisor graph $Γ(\mathcal L)$ and determine some properties of this subgraph in terms of some lattice theoretic objects such as associated prime ideals of $\mathcal L$.