AI 中文总结
该研究探讨正交模格上近似同态的稳定性，分析分离性的适用条件，开发布尔块上的分块稳定性理论，给出粘合准则及有限维投影格的分块逼近结果。
AI 中文摘要
由Ulam发起的稳定性纲领研究代数恒等式的近似解何时必须接近精确解。对于格而言，这引出了如下问题：当一个映射几乎保持并和交运算时，它何时能被一个真正的格同态逼近。Badora、Kochanek和Przebieracz针对分配格开发了基于邻域的框架，核心是分离（三明治）引理，该引理通过序包络在次并同态和超并同态之间构造出一个精确的并同态。我们重新审视这一机制，识别出使用分配性的唯一一步：即对位于某个并之下的元素的分解恒等式。若没有分配性，分离性在模格$M_3$和一个小型有限正交模格中就会失效。在正面结果方面，我们证明当下界映射是保序的时，分离性在任意格中都成立。对于正交模格——量子逻辑的代数模型——我们在布尔块上开发了分块稳定性理论。相容对上的近似恒等式会在每个块上产生精确的同态选择（针对并、针对交，以及在双容许性下针对两种运算）。我们给出了若干用于组装分块选择的粘合准则，并提供了一个具体的有限例子，表明当块重叠非平凡时粘合会失效。最后，遵循Kalton-Roberts的思路，我们通过有限可加测度获得正交模格上几乎可加函数的分块逼近结果，并在有限维投影格上给出了示例。
英文摘要
The stability programme initiated by Ulam asks when approximate solutions to algebraic identities must lie near exact ones. For lattices, this leads to the question of when a map that nearly preserves joins and meets can be approximated by a genuine lattice homomorphism. Badora--Kochanek--Przebieracz developed a neighbourhood-based framework for distributive lattices, centred on a separation (sandwich) lemma that constructs an exact join homomorphism between a join-subhomomorphism and a join-superhomomorphism via an order envelope. We revisit this mechanism and identify the single step at which distributivity is used: a decomposition identity for elements lying below a join. Without distributivity, separation can fail already in the modular lattice $M_3$ and in a small finite orthomodular lattice. On the positive side, we show that separation holds in arbitrary lattices whenever the lower bounding map is isotone. For orthomodular lattices---algebraic models of quantum logic---we develop a blockwise stability theory on Boolean blocks. Approximate identities on compatible pairs yield exact homomorphic selections on each block (for joins, for meets, and for both operations under bi-admissibility). We present several gluing criteria for assembling blockwise selections, and we give a concrete finite example showing that gluing can fail when block overlaps are non-trivial. Finally, in the spirit of Kalton--Roberts, we obtain blockwise approximation results for nearly additive functions on orthomodular lattices by finitely additive measures, with an illustration on finite-dimensional projection lattices.
Comments16 pp., accepted for publication in Notre Dame Journal of Formal Logic
