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The aim of this paper is to introduce the notion of fantastic deductivesystems on generalizations of fuzzy structures, and to emphasize their role inthe probability theory on these algebras. We give a characterization ofcommutative pseudo-BE algebras and we generalize an axiom system consisting offour identities to the case of commutative pseudo-BE algebras. We define thefantastic deductive systems of pseudo-BE algebras and we investigate theirproperties. It is proved that, if a pseudo-BE(A) algebra $A$ is commutative,then all deductive systems of $A$ are fantastic. Moreover, we generalize thenotions of measures, state-measures and measure-morphisms to the case ofpseudo-BE algebras and we also prove that there is a one-to-one correspondencebetween the set of all Bosbach states on a bounded pseudo-BE algebra and theset of its state-measures. The notions of internal states and state-morphismoperators on pseudo-BCK algebras are extended to the case of pseudo-BE algebrasand we also prove that any type II state operator on a pseudo-BE algebra is astate-morphism operator on it. The notions of pseudo-valuation and commutativepseudo-valuation on pseudo-BE algebras are defined and investigated. For thecase of commutative pseudo-BE algebras we prove that the two kind ofpseudo-valuations coincide. Characterizations of pseudo-valuations andcommutative pseudo-valuations are given. We show that the kernel of a Bosbachstate (state-morphism, measure, type II state operator, pseudo-valuation) is afantastic deductive system.

Fantastic deductive systems in probability theory on generalizations of fuzzy structures