Irreflexive modality, the Dummett logic, and continual chains

We construct a countable family of extensions of the logic of finite chains (the Dummett logic) in the language containing the standard logical connectives and a new connective (irreflexive modality), each of which determines in the Dummett logic a new logical connective in the sense of Novikov. Two arbitrary logics on this list are incompatible over the Dummett logic; i.e., their union contains a formula absent from the Dummett logic.