We prove that the variety of nuclear implicative semilattices is locally finite, thus generalizing Diego's Theorem. The key ingredients of our proof include the coloring technique and construction of universal models from modal logic. For this we develop duality theory for finite nuclear implicative semilattices, generalizing K\"ohler duality. We prove that our main result remains true for bounded nuclear implicative semilattices, give an alternative proof of Diego's Theorem, and provide an explicit description of the free cyclic nuclear implicative semilattice.

中文翻译：

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核蕴涵半格的迭戈定理

我们证明核蕴涵半格的种类是局部有限的，从而推广了迭戈定理。我们证明的关键要素包括着色技术和从模态逻辑构建通用模型。为此，我们开发了有限核蕴涵半格的对偶理论，推广了 K\"ohler 对偶性。我们证明了我们的主要结果对于有界核蕴涵半格仍然成立，给出了迭戈定理的替代证明，并提供了自由循环的明确描述核蕴涵半格。