We study many-valued coalgebraic logics with semi-primal algebras of truth-degrees. We provide a systematic way to lift endofunctors defined on the variety of Boolean algebras to endofunctors on the variety generated by a semi-primal algebra. We show that this can be extended to a technique to lift classical coalgebraic logics to many-valued ones, and that (one-step) completeness and expressivity are preserved under this lifting. For specific classes of endofunctors, we also describe how to obtain an axiomatization of the lifted many-valued logic directly from an axiomatization of the original classical one. In particular, we apply all of these techniques to classical modal logic.

翻译：我们研究以半原始真值代数为基础的多值余代数逻辑。我们提供了一种系统方法，将定义在布尔代数簇上的端函子提升到由半原始代数生成的簇上。我们证明，这可以扩展为一种将经典余代数逻辑提升为多值逻辑的技术，并且（单步）完备性和表达性在此提升下得以保持。针对特定类别的端函子，我们还描述了如何直接从原始经典逻辑的公理化中获得提升后多值逻辑的公理系统。特别地，我们将所有这些技术应用于经典模态逻辑。