Irreflexive modality on a chain of type $\omega$ and Novikov completeness

We consider a $\varphi$-logic $\mathcal{L}(\omega)$ of a frame of order type $\omega$ endowed with an irreflexive operator. The irreflexive modality in $LC$ was treated by the author in [Sib. Mat. Zh., 55, No. 1 (2014), 228–234] 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 $\omega$, on the other hand, generates inconsistent $\varphi$-logics over $LC$. There, also, it was stated that $\mathcal{L}(\omega)$ defines a new nonconstant connective in $LC$. Here we establish that the $\varphi$-logic $\mathcal{L}(\omega)$ is Novikov complete over $LC$.