We show that all finite lattices, including non-distributive lattices, arise as stable matching lattices when all agents have path-independent choice functions. This result answers an open question of Blair~\cite{blair1988lattice}. In the process, we introduce new tools to reason on general lattices for optimization purposes: the \emph{partial representation} of a lattice, which partially extends Birkhoff's representation theorem to non-distributive lattices; the \emph{distributive closure} of a lattice, which gives such a partial representation; and \emph{join constraints}, which can be added to the distributive closure to obtain a representation for the original lattice. Then, we use these techniques to show that the minimum cost stable matching problem under the same standard assumptions on choice functions is NP-hard, by establishing a connection with antimatroid theory.

翻译：我们证明，当所有代理都拥有路径独立选择函数时，所有有限格（包括非分配格）均可作为稳定匹配格出现。这一结果回答了Blair~\cite{blair1988lattice}提出的一个开放问题。在此过程中，我们引入了用于优化目的的新工具来推理一般格：格的\emph{部分表示}，该表示将Birkhoff表示定理部分扩展至非分配格；格的\emph{分配闭包}，它给出了这样一种部分表示；以及\emph{并约束}，可将其添加至分配闭包以获得原始格的表示。随后，我们利用这些技术，通过与反拟阵理论建立联系，证明在关于选择函数的相同标准假设下，最小成本稳定匹配问题是NP难的。