State BE-algebras

Abstract

Flaminio and Montagna were the first to present a unified approach to states and probabilistic many-valued logic in a logical and algebraic setting (Flaminio and Montagna, Inter J Approx Reason, 50:138–152, 2009, [95]). They added a unary operation, called internal state or state operator to the language of MV-algebras which preserves the usual properties of states. A more powerful type of logic can be given by algebraic structures with internal states, and they are also very interesting varieties of universal algebras. Di Nola and Dvurecenskij introduced the notion of a state-morphism MV-algebra which is a stronger variation of a state MV-algebra (Di Nola and Dvurecenskij, Ann Pure Appl Logic 161:161–173, 2009, [65]). The notion of a state operator was extended by Rachunek and Salounova in (Soft Comput 15:327–334, 2011, [196]) for the case of GMV-algebras (pseudo MV-algebras). State operators and state-morphism operators on BL-algebras were introduced and investigated in Ciungu et al. (Soft Comput 15:619–634, 2011, [57]), and subdirectly irreducible state-morphism BL-algebras were studied in Dvurecenskij (Arch Math Logic 50:145–160, 2011, [89]). Recently, the state BCK-algebras and state-morphism BCK-algebras were defined and studied in Borzooei et al. (State BCK-algebras and state-morphism BCK-algebras, 2013, [22]).