States and topologies on residuated lattices ∗

Multiple-valued logics are non-classical logics. They are similar to classical logic because they accept the principle of truth-functionality, namely, the truth of a compound sentence is determined by the truth values of its component sentences (and so remains unaffected when one of its component sentences is replaced by another sentence with the same truth value). But they differ from classical logic by the fundamental fact that they do not restrict the number of truth values to only two: they allow a larger set of truth degrees. The axiomatization of probability was done by Kolmogorov in 1933 and both probability and statistics had developed into major fields. But new areas of science have appeared during the last century, such as quantum mechanics, which do not satisfy the Kolmogorov axioms. These new fields of science require a probability theory based on non-classical logics. In analogy to probability measure, the states on multiple-valued algebras proved to be the most suitable models for averaging the truth-value in their corresponding logics. Continuous states play an important role for the development of these models and they are closely connected to the concepts of converge in multiple-valued logic algebras. The study of states on MV-algebras is a very actual problem which arises from the general problem of investigating the probabilities defined for logical systems. States on an MV-algebra (A,⊕,∗ , 0) were first introduced by F. Kôpka and F. Chovanec in [56] and by D. Mundici in [61] as a functions s : A → [0, 1] satisfying the conditions: s(1) = 1 (normality), s(x⊕ y) = s(x) + s(y) if x ̄ y = 0 (additivity), where x ̄ y = (x∗ ⊕ y∗)∗. They are analogous to the finitely additive probability measures on Boolean algebras and play a crucial role in MV algebraic probability theory [72]. In [36], A. Dvurečenskij defined the states on a pseudo MV-algebra in essentially the same way as for MV algebras. In a very similar way as for MV algebras, B. Riečan gave the definition of states on a BL algebra [70]. In [43], the notion of Bosbach state for pseudo BL algebras is defined by using an identity studied by Bosbach in residuation groupoids [6]. For a good pseudo BL-algebra, the Riečan states were defined in [43] to extend the additive measures introduced by Riečan for BL algebras in [70] and it was proved that every Bosbach state is a Riečan state, but the converse is an open question. In [43], it was also proved that the existence of a state on a pseudo BL algebra is equivalent with