Consider points $(x_1,y_1)$ and $(x_2, y_2)$ in $\Re^2$. We say $(x_1, y_1)$ is {\em dominated} by $(x_2,y_2)$ if $x_1 < x_2$ and $y_1 < y_2$. If $(x_1, y_1)$ is dominated by $(x_2,y_2)$, then we write $(x_1,y_1) <_D (x_2,y_2)$. For any finite subset $P \subseteq \Re^2$, the partially ordered set $(P, \le_D)$ is called the dominance poset. In this paper we present two distributed algorithms to compute the maximal points of a dominance poset, on a linear array of machines. Our first algorithm to compute the maximal points of a dominance poset begins with each point initially believing that it is a maximal point. As the algorithm proceeds each point that is not maximal discovers a point that dominates it. Our second algorithm is self-stabilizing in the sense that no particular initial state is assumed. The machines start from arbitrary initial states and converge to a final state in which all maximal points are identified. The self-stabilizing nature of our second algorithm makes it tolerant to transient faults.