THE ANNIHILATOR GRAPH OF A 0-DISTRIBUTIVE LATTICE

In this article, for a lattice L, we define and investigate the annihilator graph ag(L) of L which contains the zero-divisor graph of L as a subgraph. Also, for a 0-distributive lattice L, we study some properties of this graph such as regularity, connectedness, the diameter, the girth and its domination number. Moreover, for a distributive lattice L with Z(L) not equal {0}, we show that ag(L) = Gamma (L) if and only if L has exactly two minimal prime ideals. Among other things, we consider the annihilator graph ag(L) of the lattice L = (D(n),vertical bar) containing all positive divisors of a non-prime natural number n and we compute some invariants such as the domination number, the clique number and the chromatic number of this graph. Also, for this lattice we investigate some special cases in which ag(D(n)) or are planar, Eulerian or Hamiltonian.