Partitioning Kripke frames of finite height

In this paper we prove the finite model property and decidability of a family of modal logics. A binary relation $R$ is said to be pretransitive if $R^*=\bigcup_{i\leqslant m} R^i$ for some $m\geqslant 0$, where $R^*$ is the transitive reflexive closure of $R$. By the height of a frame $(W,R)$ we mean the height of the preorder $(W,R^*)$. We construct special partitions (filtrations) of pretransitive frames of finite height, which implies the finite model property and decidability of their modal logics.