**Authors:** Ratsa M. F.,

Rusu Andrei
### Abstract

We consider the diagonalizable algebra $\frak M = <\!M;\, \Omega\!>$, $\Omega = \{\&,\; \lor,\; \supset,\; \neg, \;\Delta\}$ of the set of all infinite binary sequences of the type $\alpha = (\mu_1,\; \mu_2, \;\dots)$, $\mu_i\in\{0,\;1\}$, $i = 1, \;2, \;\dots$. The Boolean operations $\&, \;\lor, \;\supset, \;\neg\;$ over elements of $M$ are defined componentwise, and the operation $\Delta$ over element $\alpha$ we define by the equality $\Delta\alpha = (1, \;\nu_1, \;\nu_2, \;\dots )$, where $\nu_i = \mu_1\/\&\cdots\&\/\mu_i$. We consider then the subalgebra $\frak M^*$
of the algebra $\frak M$ which is generated by its zero element $(0, \;0, \;\dots )$. Let the logic $L$, which is an extension of the provability logic $GL$, satisfies the relations: $GL\subseteq L\subseteq L\frak M^*$.
Then the classes $K_0$, \;$K_1$, \;$K_2$, \dots of formulae, which preserve on the algebra $\frak M^*$
respectively the predicates $\square x = 0$, \;$\square x\le\Delta 0$, \;$\square x\le\Delta^2 0$, \dots,
constitute a numerable collection of precomplete in the logic $L$ classes of formulae.

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