Suzumura consistency, an alternative approach

Suzumura consistency is known as a sufficient and necessary condition for a binary relation to have an order extension. We advocate the use of equivalent but negation-free forms of Suzumura consistency and of the related notion of compatible extension. From a methodological perspective, our proposals make possible to work more abstractly, in the algebra of relations, and to give more direct proofs. To illustrate this we reconsider various forms and proofs of the order extension principle. As a complement we adopt to quasi-orders J.L. Bell’s argument that Gödel–Dummett logic is necessary for order extension.