This note aims to clarify the relations between three ways of constructing complete lattices that appear in three different areas: (1) using ordered structures, as in set-theoretic forcing, or doubly ordered structures, as in a recent semantics for intuitionistic logic; (2) using compatibility relations, as in semantics for quantum logic based on ortholattices; (3) using Birkhoff’s polarities, as in formal concept analysis.

中文翻译：

---

完成晶格的三条路：顺序，兼容性，极性

本注释旨在阐明在三个不同区域中出现的构造完整晶格的三种方式之间的关系：（1）使用有序结构（如集合论强制）或双重有序结构（如最近针对直觉逻辑的语义）；（2）使用兼容性关系，例如基于正交晶格的量子逻辑的语义；（3）使用伯克霍夫（Birkhoff）的两极性，就像在形式概念分析中一样。