A Modal Logic That is Complete with Respect to Strictly Linearly Ordered $A$-Models

An axiomatization is furnished for a polymodal logic of strictly linearly ordered $A$-frames: for frames of this kind, we consider a language of polymodal logic with two modal operators, $\Box_<$ and $\Box_\prec$. In the language, along with the operators, we introduce a constant $\beta$, which describes a basis subset. In the language with the two modal operators and constant $\beta$, an $L\alpha$-calculus is constructed. It is proved that such is complete w. r. t the class of all strictly linearly ordered $A$-frames. Moreover, it turns out that the calculus in question possesses the finite-model property and, consequently, is decidable.