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We define the concepts of minimal p-morphic image and basic p-morphism for transitive Kripke frames. These concepts are used to determine effectively the least number of variables necessary to axiomatize a tabular extension of K4, and to describe the covers and co-covers of such a logic in the lattice of the extensions of K4. [ABSTRACT FROM AUTHOR]

In this paper we show the adequacy of tense logic with unary operators for dealing with finite trees. We prove that models on finite trees can be characterized by tense formulas, and describe an effective method to find an axiomatization of the theory of a given finite tree in tense logic. The strength of the characterization is shown by proving that adding the binary operators "Until" and "Since" to the language does not result in a better description than that given by unary tense logic; although the greater expressive power of "Until" and "Since" can be exploited by using the semantics of e-frames instead of traditional Kripke semantics. [ABSTRACT FROM AUTHOR]

We find a short way to construct a formula which axiomatizes a given finite frame of the modal logic K, in the sense that for each finite frame A, we construct a formula ω A which holds in those and only those frames in which every formula true in A holds. To obtain this result we find, for each finite model $$\mathfrak{A}$$ and each natural number n, a formula ω $$\mathfrak{A}$$ which holds in those and only those models in which every formula true in $$\mathfrak{A}$$ , and involving the first n propositional letters, holds. [ABSTRACT FROM AUTHOR]

We give a representation of finitely generated free interior algebras and of the Heyting algebras of their open elements. By means of the representation we give a new and strengthened version of some earlier theorems of McKinsey-Tarski and Blok, etc. regarding such algebras. [ABSTRACT FROM AUTHOR]