发表机构
University of Malta（马耳他大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究无穷分配格的正则子格在完备化下的行为，通过O闭理想构造完备格，比较Dedekind-MacNeille完备化，并证明三维是有限实链乘积的有界子格具有非分配完备化的最小维度。
AI 中文摘要
我们研究了无穷分配格的正则子格在完备化下的行为。利用序收敛，我们将无穷分配格\\(L\\)与\\(O\\)-闭序理想的完备格\\(\mathfrak I_L\\)联系起来，并描述了相应的闭包算子\\(A\mapsto A^{\sigma_L}\\)。我们证明了每个格同态都会在相应的\\(O\\)-闭理想格之间诱导出一个典范映射，并将\\(\mathfrak I_L\\)与Dedekind--MacNeille完备化\\(\DM(L)\\)进行比较。当\\(\DM(L)\\)仍为无穷分配时，这一比较产生了格同态的扩张结果，以及\\(\DM(Y)\\)在环境完备格内的并正则实现。我们还确定了一个尖锐的有限维障碍。我们构造了一个无穷分配正则子格\\(L\subseteq\mathbb R^3\\)，其Dedekind--MacNeille完备化甚至不是模格，因此无法实现为\\(\mathbb R^3\\)的子格。相比之下，我们证明了两个链的乘积的每个有界子格的Dedekind--MacNeille完备化都是分配的。因此，三维是有限个实链乘积的有界子格可能具有非分配Dedekind--MacNeille完备化的最小维度。
英文摘要
We study the behaviour of regular sublattices of infinitely distributive lattices under completion. Using order convergence, we associate with an infinitely distributive lattice \(L\) the complete lattice \(\mathfrak I_L\) of \(O\)-closed order ideals and describe the corresponding closure operator \(A\mapsto A^{σ_L}\). We show that every lattice homomorphism induces a canonical map between the corresponding lattices of \(O\)-closed ideals, and compare \(\mathfrak I_L\) with the Dedekind--MacNeille completion \(\DM(L)\). When \(\DM(L)\) remains infinitely distributive, this comparison yields extension results for lattice homomorphisms and a join-regular realization of \(\DM(Y)\) inside an ambient complete lattice. We also determine a sharp finite-dimensional obstruction. We construct an infinitely distributive regular sublattice \(L\subseteq\mathbb R^3\) whose Dedekind--MacNeille completion is not even modular, and therefore cannot be realized as a sublattice of \(\mathbb R^3\). In contrast, we prove that the Dedekind--MacNeille completion of every bounded sublattice of a product of two chains is distributive. Thus dimension three is the least dimension in which a bounded sublattice of a finite product of real chains can have a non-distributive Dedekind--MacNeille completion.
Comments14 pages
