Generalized Kripke semantics for Nelson's logic

A completeness theorem for logics $N4^N$ and $N3^0$ is proved. A characterization by classes of $N4^N$- and $N3^0$-models is presented, and it is proved that all logics of four types $\eta(L)$, $\eta^3(L)$, $\eta^n(L)$, and $\eta^0(L)$ are Kripke complete iff so are their respective intuitionistic fragments $L$. A generalized Kripke semantics is introduced, and it is stated that such is equivalent to an algebraic semantics. The concept of a $p$-morphism between generalized frames is defined and basic statements on $p$-morphisms are proved.