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.