On Beck’s coloring of posets

We study Beck-like coloring of partially ordered sets (posets) with a least element 0. To any poset PP with 0 we assign a graph (called a zero-divisor graph) whose vertices are labelled by the elements of PP with two vertices x,yx,y adjacent if 0 is the only element lying below xx and yy. We prove that for such graphs, the chromatic number and the clique number coincide. Also, we give a condition under which posets are not finitely colorable.