On $S5$-$T$-$Y$ logic

The least $S5$-$T$-$Y$ logic is studied. The language of this logic is formed by addition of the connectives $T$ (‘tomorrow’) and $Y$ (‘yesterday’) to the language of $S5$. The $T$-$Y$ logic axioms (cf. [2]) and the axioms
\begin{align*} &\square A\to TA\land YA,\quad TA\lor YA\to\diamond A \\ &T\square A\leftrightarrow \square A,\quad Y\square A\leftrightarrow \square A \end{align*}
with the rule of the substitution are added to the axiomatic of $S5$. It is proved, that $L_\infty$ is determined of the Kripke frame which is the union of the frames which has the order type $Z$ (the set of the integers). It is used the reducing of the formulas to perfect disjunctive normal forms (PDNF).