States of finite GBL-algebras with monoidal sum

Generalized BL-algebras, i.e. divisible residuated lattices, provide the semantics for a generalization of Basic Logic where the axiom of prelinearity does not hold. Informally, GBL-algebras generalize Heyting algebras in a similar way as MV-algebras generalize Boolean algebras. We introduce the operation of sum in finite GBL-algebras and we axiomatize the obtained finite structures, called GBL⊕-algebras. We hence define states of GBL⊕-algebras, extending MV-algebraic states, and we prove that they are determined by their restriction on the Heyting skeleton. Extremal states are also characterized in terms of densities concentrated in a unique join-prime idempotent.