Irreflexive Modality on a Chain of Type ω and P. S. Novikov Completeness

We consider a φ-logic L(ω) of a frame of order type ω endowed with an irreflexive operator. The irreflexive modality in LC was treated by the author in [Sib. Math. J., 55, No. 1, 185-190 (2014)] where it was shown that this modality on the class of finite chains, on the one hand, and on a single chain of order type ω, on the other hand, generates inconsistent φ-logics over LC. There, also, it was stated that L(ω) defines a new nonconstant connective in LC. Here we establish that the φ-logic L(ω) is Novikov complete over LC.