  We propose a new definition of the representation theorem for many-valued
logics, with modal operators as well, and define the stronger relationship
between algebraic models of a given logic and relational structures used to
define the Kripke possible-world semantics for it. Such a new framework offers
a new semantics for many-valued logics based on the truth-invariance
entailment. Consequently, it is substantially different from current
definitions based on a matrix with a designated subset of logic values, used
for the satisfaction relation, often difficult to fix. In the case when the
many-valued modal logics are based on the set of truth-values that are complete
distributive lattices we obtain a compact autoreferential Kripke-style
canonical representation. The Kripke-style semantics for this subclass of modal
logics have the joint-irreducible subset of the carrier set of many-valued
algebras as set of possible worlds. A significant member of this subclass is
the paraconsistent fuzzy logic extended by new logic values in order to also
deal with incomplete and inconsistent information. This new theory is applied
for the case of autoepistemic intuitionistic many-valued logic, based on
Belnap's 4-valued bilattice, as a minimal extension of classical logic used to
manage incomplete and inconsistent information as well.
