Kripke semantics for the logic of centered Kleene algebras with intuitionistic negation

This paper introduces a Kripke-style semantics for the $1$-assertional logic of centered Kleene algebras with intuitionistic negation (KANc-algebras). The semantics is derived from Kalman's construction and interprets each formula by a pair of disjoint increasing sets, representing verification and falsification. We explicitly define the algebraic consequence relation determined by the variety of centered KAN-algebras and prove that it coincides with the global consequence relation determined by the proposed Kripke models. The proof uses the prime-filter representation of distributive p-algebras together with Kalman's construction. Building on this semantics, we develop a sound and complete analytic tableau proof system. The calculus is proved sound and complete for KANc-tautologies; no claim of termination or decidability is made here. This work establishes a relational semantics and proof theory for KANc-logic and provides a basis for future extensions.