Fantastic Deductive Systems in Probability Theory on Generalizations of Fuzzy Structures

The aim of this paper is to introduce the notion of fantastic deductive systems on generalizations of fuzzy structures, and to emphasize their role in the probability theory on these algebras. We give a characterization of commutative pseudo-BE algebras and we generalize an axiom system consisting of four identities to the case of commutative pseudo-BE algebras. We define the fantastic deductive systems of pseudo-BE algebras and we investigate their properties. It is proved that, if a pseudo-BE(A) algebra A is commutative, then all deductive systems of A are fantastic. Moreover, we generalize the notions of measures, state-measures and measure-morphisms to the case of pseudo-BE algebras and we also prove that there is a one-to-one correspondence between the set of all Bosbach states on a bounded pseudo-BE algebra and the set of its state-measures. The notions of internal states and state-morphism operators on pseudo-BCK algebras are extended to the case of pseudo-BE algebras and we also prove that any type II state operator on a pseudo-BE algebra is a statemorphism operator on it. The notions of pseudo-valuation and commutative pseudo-valuation on pseudo-BE algebras are defined and investigated. For the case of commutative pseudoBE algebras we prove that the two kind of pseudo-valuations coincide. Characterizations of pseudo-valuations and commutative pseudo-valuations are given. We show that the kernel of a Bosbach state (state-morphism, measure, type II state operator, pseudo-valuation) is a fantastic deductive system.