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

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 ۰-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.

کلیدواژه ها:

Distributive lattice ، Goldie dimension ، compressed zero-divisor graph ، domination number