Product of Partially Ordered Sets (Posets), with Potential Applications to Uncertainty Logic and Space-Time Geometry

One of the main objectives of science and engineering is to help people select the most beneficial decisions. To make these decisions, • we must know people’s preferences, • we must have the information about different events – possible consequences of different decisions, and • since information is never absolutely accurate and precise, we must also have information about the degree of certainty. All these types of information naturally lead to partial orders: • For preferences, a ≺ b means that b is preferable to a. This relation is used in decision theory. • For events, a ≺ b means that a can influence b. This causality relation is used in space-time physics. • For uncertain statements, a ≺ b means that a is less certain than b. This relation is used in logics describing uncertainty such as fuzzy logic. In many practical situations, we are analyzing a complex system that consists of several subsystems. Each subsystem can be described as a separate ordered space. To get a description of the system as a whole, we must therefore combine these ordered spaces into a single space that describes the whole system. In this paper, we consider the general problem of how to combine two ordered spaces A1 and A2 into one. We also analyze which properties of orders are preserved under the resulting products.