Rankability as a Strong Form of Coherence

Abstract

A preference relation is a binary endorelation B over an arbitrary ground set X of alternatives. The condition that B is a strict partial order (or, equivalently that B is transitive and asymmetric) is often viewed as an expression of the coherence of the preference relation; the stronger condition that B is a strict weak order (or, equivalently that B is negatively transitive and asymmetric) is often viewed as an expression of the rankability of the alternatives in X. As is known, the requisite that a relation B on a ground set X is a strict partial order is equivalent to the assumption that for every pair (y,x) in X×X the implication y∈B(x)⇒B(y)⊂B(x) is satisfied (with B(t) denoting {s∈X:(s,t)∈B} for all t∈X and with ⊂ denoting strict inclusion). The paper introduces the new notions of a strongly coherent relation and that of a completely coherent relation: a strongly coherent relation is defined as one that satisfies the previous one-way implication as a double implication; a completely coherent relation is defined as one whose restriction to any subset of its ground set is strongly coherent. Three are the principal results of the paper that justify its title: (1) any relation on a finite ground set is a strict weak order if and only if it is strongly coherent; (2) any Ferrers relation is a strict weak order if and only if it is strongly coherent; (3) any relation is a strict weak order if and only if it is completely coherent.