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

In this paper, for a distributive lattice L, we study and compare some lattice theoretic features of L and topological properties of the Stone spaces Spec(L) and Max(L) with the corresponding graph theoretical aspects of the zero-divisor graph Γ(L). Among other things, we show that the Goldie dimension of L is equal to the cellularity of the topological space Spec(L) which is also equal to the clique number of the zero-divisor graph Γ(L). Moreover, the domination number of Γ(L) will be compared with the density and the weight of the topological space Spec(L). For a 0-distributive lattice L, we investigate the compressed subgraph ΓE(L) of the zero-divisor graph Γ(L) and determine some properties of this subgraph in terms of some lattice theoretic objects such as associated prime ideals of L.