Existence of closed and complete extensions applied to convex, homothetic an monotonic orderings

Many theories of consumer demand impose specific properties on the preference relations, e.g. convexity, monotonicity or homotheticity. Existing nonparametric tests which allow us to single out the preference relations that do not satisfy these properties are only valid in very specific contexts. This paper is an attempt to address this lacuna in the literature. We provide a theorem on the existence of complete binary extensions that satisfy properties which are closed under intersection. From this theorem we derive necessary and su_cient conditions for the existence of convex, homothetic and monotonic orderings on general domains.