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：我们研究了基于半素真值代数结构的多值余代数逻辑。本文提供了一种系统方法，可将定义在布尔代数变体上的自函子提升为由半素代数生成的变体上的自函子。我们证明该提升过程可扩展为一种将经典余代数逻辑转化为多值逻辑的技术，并且在该提升下（单步）完备性与表达性保持成立。针对特定类型的自函子，我们还描述了如何直接从原始经典逻辑的公理化出发，直接获得提升后多值逻辑的公理化方案。特别地，我们将所有这些技术应用于经典模态逻辑。