Stone 预有限性、网状谱与剩余格的预有限完备化障碍
Stone profiniteness, reticulation spectra, and profinite-completion obstructions for residuated lattices
- School of Mathematical Sciences, Guangxi Minzu University（广西民族大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文为剩余格建立预有限理论，证明 Stone-预有限定理、解决 Grätzer 型谱表示问题，并给出预有限完备化的判据与障碍定理，构造了非幂等的非完备化例子。
AI中文摘要：
我们围绕三个相关问题，为有界交换整剩余格发展了一套预有限理论。首先，我们证明了一个 Stone–预有限定理：一个拓扑剩余格是预有限的，当且仅当其底层空间是 Stone 空间。该证明通过 1 类滤子，将半环归约的闭开同余细化为完整剩余格签名中的闭开同余。其次，我们解决了类别的 Grätzer 型谱表示问题：对于剩余格的每个类 K，K 可表示的素滤子谱恰好是网状像 Ret(K) 中有界分配格的 Priestley 谱。这一归约是精确的，并且我们为若干自然的有限类和逻辑类计算了 Ret(K)，包括有限 MTL、BL、MV 以及 n-幂等/EDPC 簇。第三，我们研究预有限完备化。我们给出了一个基于稠密子代数和闭开同余的内在判据，证明了在 n-幂等簇中有限指标同余的有限生成条件下的正完备化定理，并建立了一个有限标签障碍定理。特别地，利用 Bezhanishvili–Bezhanishvili–Moraschini–Stronkowski 的 Heyting 非完备化例子以及三元素 Łukasiewicz 链 L3，我们获得了真正非幂等的预有限剩余格，它们不是预有限完备化。
英文摘要：
We develop a profinite theory for bounded commutative integral residuated lattices around three related questions. First, we prove a Stone--profinite theorem: a topological residuated lattice is profinite exactly when its underlying space is Stone. The proof refines clopen congruences of the semiring reduct into clopen congruences in the full residuated-lattice signature via the 1-class filter. Second, we solve the classwise Grätzer-type spectral representation problem: for every class K of residuated lattices, the K-representable prime-filter spectra are precisely the Priestley spectra of the bounded distributive lattices in the reticulation image Ret(K). This reduction is sharp, and we compute Ret(K) for several natural finite and logical classes, including finite MTL-, BL-, MV-, and n-potent/EDPC varieties. Third, we study profinite completions. We give an intrinsic criterion in terms of dense subalgebras and clopen congruences, prove a positive completion theorem under finite generation of finite-index congruences in n-potent varieties, and establish a finite-tag obstruction theorem. In particular, using the Heyting non-completion examples of Bezhanishvili--Bezhanishvili--Moraschini--Stronkowski and the three-element Łukasiewicz chain L3, we obtain genuinely non-idempotent profinite residuated lattices which are not profinite completions.
