It is known that Craig interpolation theorem does not hold for LTL. In this paper, Craig interpolation theo- rems are shown for some fragments and extensions of LTL. These theorems are simply proved based on an embedding-based proof method with Gentzen-type sequent calculi. Maksimova separation theorems (Maksimova principle of variable separation) are also shown for these LTL variants.

In this note we prove that a Tarski algebra is determined by the monoid of its endomorphisms as well as by the lattice of its subalgebras.

The present note oers an axiomatization for an expansion of Nelson's logic motivated by Heinrich Wansing which serves as a base logic for the framework of nonmonotonic reasoning considered by Dov Gabbay and Raymond Turner. We also show that the expansion of Wansing is not conservative  intuitionistic logic, but at least as strong as Jankov's logic.

After defining continuous extensions of binary relations on the set N of natural numbers to its Stone-ˇCech compactification βN, we establish some results about one of such extensions. This provides us with one possible divisibility relation on βN, | , and we introduce a few more, defined in a natural way. For some of them we find equivalent conditions for divisibility. Finally, we mention a few facts about prime and irreducible elements of (βN,·). The motivation behind all this is to try to translate problems in elementary number theory into βN

Generalizing ordinary topological and pretopological spaces, we introduce the notion of peritopology where neighborhoods of a point need not contain that point, and some points might even have an empty neighborhood. We brie y describe various intrinsic aspects of this notion. Applied to modal logic, it gives rise to peritopological models, a generalization of topo- logical models, a spacial case of neighborhood semantics. A new cladding for bisimulation is presented. The concept of Alexandro peritopology is used in order to determine the logic of all peritopo- logical spaces, and we prove that the minimal logic K is strongly complete with respect to the class of all peritopological spaces. We also show that the classes of T0, T1 and T2-peritopological spaces are not modal denable, and that D is the logic of all proper peritopological spaces. Finally, among our conclusions, we show that the question whether T0, T1 peritopological spaces are modal denable in H(@) remains open.

We introduce the notion of an everywhere strongly logifiable algebra: a finite non-trivial algebra A such that for every F 2 P(A) r f;;Ag the logic determined by the matrix hA; Fi is a strongly algebraizable logic with equivalent algebraic semantics the variety generated by A. Then we show that everywhere strongly logifiable algebras belong to the field of universal algebra as well as to the one of logic by characterizing them as the finite non-trivial simple algebras that are constantive and generate a congruence distributive and n-permutable variety for some n > 2. This result sets everywhere strongly logifiable algebras surprisingly close to primal algebras. Nevertheless we shall provide examples everywhere strongly logifiable algebras that are not primal. Finally, some conclusion on the problem of determining whether the equivalent algebraic semantics of an algebraizable logic is a variety is obtained.