Glivenko Classes of Sequents for Temporal Logic with Time Gaps

Let LB be a sequent calculus of the first-order classical temporal logic TB with time gaps. Let, further, LBJ be the intuitionistic counterpart of LB. In this paper, we consider conditions under which a sequent is derivable in the calculus LBJ if and only if it is derivable in the calculus LB. Such conditions are defined for sequents with one formula in the succedent (purely Glivenko -classes) and for sequents with the empty succedent (Glivenko -classes).     

剩余格的模糊滤子理论

运用模糊集的方法和原理进一步深入研究剩余格的滤子问题.在剩余格中引入了模糊预线性滤子,模糊可除滤子和模糊Glivenko滤子三类新的模糊滤子概念,给出了它们的若干性质和等价刻画.系统讨论了这三类模糊滤子以及模糊正关联滤子,模糊Boolean滤子,模糊MV滤子和模糊正则滤子间的相互关系,证明了一个模糊滤子为模糊MV滤子当且仅当它既是模糊正则滤子又是模糊可除滤子的结论.     

Glivenko-Cantelli type theorems: An application of the convergence theory of stochastic suprema

The uniform convergence of empirical processes on certain classes of sets follows from the convergence theory for random lower semicontinuous functions studies in the context of stochastic optimization. In the process, a richer class of sets for which one can prove this type of result is exhibited.Research supported in part by grants from Ministero Publica Istruzione and the National Science Foundation.     

Generalized Bosbach and Rie?an states based on relative negations in residuated lattices

Bosbach and Rie?an states on residuated lattices both are generalizations of probability measures on Boolean algebras. Recently, two types of generalized Bosbach states on residuated lattices were introduced by Georgescu and Mure?an through replacing the standard MV-algebra in the original definition with arbitrary residuated lattices as codomains. However, several interesting problems there remain still open. The first part of the present paper gives positive answers to these open problems. It is proved that every generalized Bosbach state of type II is of type I and the similarity Cauchy completion of a residuated lattice endowed with an order-preserving generalized Bosbach state of type I is unique up to homomorphisms preserving similarities, where the codomain of the type I state is assumed to be Cauchy-complete. Consequently, many existing results about generalized Bosbach states can be further strengthened. The second part of the paper introduces the notion of relative negation (with respect to a given element, called relative element) in residuated lattices, and then many issues with the canonical negation such as Glivenko property, semi-divisibility, generalized Rie?an state of residuated lattices can be extended to the context of such relative negations. In particular, several necessary and sufficient conditions for the set of all relatively regular elements of a residuated lattice to be special residuated lattices are given, of which one extends the well-known Glivenko theorem, and it is also proved that relatively generalized Rie?an states vanishing at the relative element are uniquely determined by their restrictions on the MV-algebra consisting of all relatively regular elements when the domain of the states is relatively semi-divisible and the codomain is involutive.     

Fuzzy Weak Regular,Strong and Preassociative Filters in Residuated Lattices

In this paper, the notions of fuzzy weak regular, strong and preassociative filters are introduced with some properties of them investigated. In particular, under the context of Glivenko algebras, fuzzy weak regular filters and regular ones are equivalent and characterizations of Glivenko algebras are concluded by characterizations of fuzzy strong filters. At last the notion of fuzzy preassociative filters are defined, which are proved to coincide with fuzzy Boolean filters, and then some new alternative definitions of Boolean algebras are obtained by this type of filters.     

Free‐decomposability in varieties of semi‐Heyting algebras

In this paper we prove that the free algebras in a subvariety $\mathcal V$ of the variety $\mathcal {SH}$ of semi‐Heyting algebras are directly decomposable if and only if $\mathcal V$ satisfies the Stone identity.