On the extension of a preorder under translation invariance

This paper proves 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. We also prove that the extension may inherit a property of continuity. As an application, we prove the existence of a complete translation-invariant strict preorder on ℝ which transgresses scalar invariance and also the existence of a complete translation-invariant preorder satisfying the social choice axioms strong Pareto and fixed--step-anonymity on a set X^{ℕ₀}, where X is a divisible commutative group. Moreover, the two extension results are used to make scalar invariance appear as a consequence of translation invariance under a continuity requirement.