A small set of axioms for residuated logic

We show that a set of three axioms, together with modus ponens, suffices to build up a formal system for the class R of all quadruples (L, ⩽, T, J), in which (L, ⩽) is a partially ordered set with a top element, and T is a triangular norm on (L, ⩽) that has a residuated implication J. We show that this R is not a variety.