The aim of this paper is to introduce some techniques that can be used in the study of stochastic processes which have as parameter set the positive quadrant of the plane R2+. We define stopping lines and derive an interesting property of measurability for them. The notion of predictability is developed, and we show the connection between predictable processes, fields associated with stopping lines, and predictable stopping lines. We also give a theorem of section for predictable sets. Extension to processes indexed by any partially ordered set with some regularity assumptions can be carried out quite easily with the same techniques.