Coherent orders

A preference relation is a binary endorelation B over an arbitrary ground set X of alternatives. The requisite that B is a strict partial order relation is equivalent to positing that the implication y ∈ B(x) ⇒ B(y) ⊂ B(x) – with B(t) denoting {s ∈ X : (s, t) ∈ B} for all t ∈ X and with ⊂ denoting strict inclusion – holds true for every pair (y, x) in X × X. The paper introduces the notion of a coherent order relation, defined as one that satisfies the previous one-way implication as a double implication. Examined are some properties of coherent order relations and how they connect with strict weak order relations and with transitive relations. In particular, the paper shows various conditions for a transitive relation to be a coherent order relation and for a coherent order relation to be a strict weak order relation.