The modal logic based on $f$-spaces is considered. The conception of $f$-space was introduced by Ershov with the purpose to develop the theory of computable functionals. If the least element exists in $X$ then $f$-space is called $f_0$-space. Frames $\langle X,X_0,R\rangle$ are considered being similar to Kripke's frames. If the frame $\langle X,X_0,\le\rangle$ is a linearly ordered $f_0$-space then the frame $\langle X,X_0,R\rangle$, where for any $x,y\in X$ $(xRy\Longleftrightarrow x\le y$ and $x\ne y)$, is called a linearly ordered $f_0$-frame. The tense modalities $\Diamond$ and $\Box$ associated with the relation $R$ are introduced. The set $X_0$ is represented as a constant $\beta$. A logic calculus $L'$ is introduced obtained by adding to the minimal calculus $K$ some axioms. The following theorems are proved: Theorem 1. In any linearly ordered $f$-frame the formulas of $L'$ are valid. Theorem 2. Every formula valid in all linearly ordered $f_0$-frames is a theorem of $L'$. Since any $f_0$-frame is a $f$-frame then $L'$ complete with respect to the class of all linearly ordered $f_0$-frames, and also with respect to the class of all linearly ordered $f$-frames.

Graduated from Faculty of Mathematics and Mechanics of Novosibirsk State University in 2000 (department of algebra and logic). Since 2000 I am post graduate student at the Institute of Mathematics of Sibirean Branch of Russian Academy of Sciences.

• Murzina V. F. The polymodal logic based on $A$-spaces // IIS SB RAS, Novosibirsk, 2000. 15p. (Preprint 73).
• Murzina V. F. The completeness theorem for modal logic based on linearly ordered $f_0$-spaces // Collequim Logicum. Abstracts of the LC 2001, v. 4, p. 137.

• Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk