We present a new method, the Subdivision Construction, for proving the finite model property (the fmp) for broad classes of modal logics and modal rule systems. The construction builds on the framework of stable canonical rules, and produces a finite modal algebra (finite modal space) that will be a finite countermodel of such rules, yielding the fmp. We apply the Subdivision Construction for proving the fmp for logics and rule systems axiomatized by stable canonical formulas and rules of finite modal algebras of finite height. We also observe that these logics and rule systems are union-splittings in corresponding lattices. As a consequence, we identify a class of union-splittings in $\mathsf{NExt}(\mathsf{K4})$ with the degree of Kripke incompleteness 1.

翻译：我们提出了一种新方法——细分构造，用于证明广泛类别的模态逻辑及模态规则系统具有有限模型性质。该构造基于稳定典范规则的框架，通过构建有限模态代数（有限模态空间）来生成此类规则的有限反模型，从而得到有限模型性质。我们应用细分构造证明了由有限高度有限模态代数的稳定典范公式与规则公理化的逻辑及规则系统具有有限模型性质。同时，我们观察到这些逻辑与规则系统在对应格中表现为并分裂。由此，我们在$\mathsf{NExt}(\mathsf{K4})$中识别出一类具有克里普不完全性度为1的并分裂。