Title: Fantastic deductive systems in probability theory on generalizations of fuzzy structures,
Abstract: 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
state-morphism 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 pseudo-BE 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.