On the Extension and Decomposition of a Preorder under Translation Invariance

We prove the existence, for a translation-invariant preorder on a divisible commutative group, of a complete preorder extending the preorder in question and satisfying translation invariance (theorem 1). We also prove that the extension may inherit a property of continuity (theorem 2). This property of continuity may lead to scalar invariance. By seeking to clarify the relationship between continuity and scalar invariance under translation invariance, we are led to formulate a theorem that asserts the existence of a continuous linear weak representation under a certain condition (theorem 3). The application of these results in a space of infinite real sequences shows that this condition is weaker than the axiom super weak Pareto, and that the latter is itself weaker than the axiom monotonicity for non-constant preorders. Thus, theorem 3 is a strengthening of theorem 4 of Mabrouk 2011. It also makes it possible to show the existence of a sequence of continuous linear preorders whose lexicographic combination constitutes the finest combination coarser than the preorder in question (theorem 4). This decomposition makes it possible to handle continuous functions instead of preoders when one looks for optima, which may be more practical. Finally we apply this decomposition to the preorder catching-up. Several examples are provided.