Nelson代数、剩余格与粗糙集：综述
Nelson algebras, residuated lattices and rough sets: A survey
AI总结：
本文综述了Nelson代数作为带强否定构造逻辑代数对应物在过去二十年的发展，探讨其推广N4-lattices及在对偶性和粗糙集理论中的应用，并给出基于粗糙集的表示定理。
摘要：
在过去50年中，Nelson代数作为带有强否定的Nelson构造逻辑的代数对应物，得到了杰出学者的广泛研究。尽管有这些研究，但目前仍缺乏对该主题的全面综述，并且Nelson代数理论对大多数逻辑学家来说仍然很大程度上是未知的。本文旨在通过聚焦该领域过去二十年中的基本发展来填补这一空白。此外，我们探讨了Nelson代数的推广，例如对应于Nelson逻辑的次协调版本的N4-lattices，以及它们在逻辑学家感兴趣的其他领域中的应用，如对偶性和粗糙集理论。一个一般的表示定理指出，每个Nelson代数都同构于由拟序导出的基于粗糙集的Nelson代数的一个子代数。此外，一个公式是Nelson逻辑的定理，当且仅当它在由拟序导出的每个有限Nelson代数中有效。
英文摘要：
Over the past 50 years, Nelson algebras have been extensively studied by distinguished scholars as the algebraic counterpart of Nelson's constructive logic with strong negation. Despite these studies, a comprehensive survey of the topic is currently lacking, and the theory of Nelson algebras remains largely unknown to most logicians. This paper aims to fill this gap by focussing on the essential developments in the field over the past two decades. Additionally, we explore generalisations of Nelson algebras, such as N4-lattices which correspond to the paraconsistent version of Nelson's logic, as well as their applications to other areas of interest to logicians, such as duality and rough set theory. A general representation theorem states that each Nelson algebra is isomorphic to a subalgebra of a rough set-based Nelson algebra induced by a quasiorder. Furthermore, a formula is a theorem of Nelson logic if and only if it is valid in every finite Nelson algebra induced by a quasiorder.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
