AI 中文总结
该研究在句法概念格中定义有限幺半群压缩数，得出伪簇的压缩高度仅为1、2或无穷的三分结论，明确了不同类型幺半群的稳定元数及相关复杂度问题。
AI 中文摘要
Clark的句法概念格（SCL）记录双侧分布结构，Wurm将其扩展至任意有限元数的元组。我们研究\\(\operatorname{cmp}_f(L)\\)，即有限幺半群观测的最小像大小，该观测通过主层上的元数\\(f\\)保护带防护的元组替换。对于正则语言，我们将\\(\operatorname{cmp}_f(L)\\)精确刻画为从带点句法幺半群出发的\\(f\\)-分离关系态射的陪域的最小基数。设\\(\operatorname{ch}(\mathbf V)\\)表示压缩数在伪簇\\(\mathbf V\\)上一致稳定的最小元数，我们的主要结果是一致高度的如下三分法：\\(\operatorname{ch}(\mathbf V)\in\{1,2,\infty\}\\)，且\\(\operatorname{ch}(\mathbf V)=\infty\\)当且仅当\\(\operatorname{Synt}(\{ab\})\in\mathbf V\\)，因此不存在有限的一致压缩高度3、4……无限情形是严格的：在\\(\langle\operatorname{Synt}(\{ab\})\rangle\\)内部，每个边界\\(d\to d+1\\)都存在无界压缩间隙，且元数层次的任意有限严格前缀均可实现。在有限侧，交换幺半群和带在元数1处稳定，每个完全正则句法幺半群在元数2处稳定；有限群核表明二元界是严格的。在一元元数处，每个非空有限简单图都可由显式长度为3的语言实现，得到精确的色数公式，且对于显式列出的长度为3的语言，判定\\(\operatorname{cmp}_1(L)\le 3\\)是NP完全问题。压缩高度1和2之间的结构边界仍待解决。
英文摘要
Clark's syntactic concept lattice (SCL) records two-sided distributional structure, and Wurm extended it to tuples of arbitrary finite arity. We study \(\operatorname{cmp}_f(L)\), the minimum image size of a finite-monoid observation that preserves guarded tuple substitution through arity \(f\) on the principal layer. For regular languages, we characterize \(\operatorname{cmp}_f(L)\) exactly as the least cardinality of the codomain of an \(f\)-separating relational morphism from the pointed syntactic monoid. Let \(\operatorname{ch}(\mathbf V)\) denote the least arity at which these compression numbers stabilize uniformly over a pseudovariety \(\mathbf V\). Our main result is the following trichotomy of possible uniform heights: \(\operatorname{ch}(\mathbf V)\in\{1,2,\infty\}\), with \(\operatorname{ch}(\mathbf V)=\infty\) if and only if \(\operatorname{Synt}(\{ab\})\in\mathbf V\). Thus no finite uniform compression height \(3,4,\ldots\) occurs. The infinite case is sharp: inside \(\langle\operatorname{Synt}(\{ab\})\rangle\), every boundary \(d\to d+1\) admits unbounded compression gaps, and arbitrary finite strict prefixes of the arity hierarchy are realizable. On the finite side, commutative monoids and bands stabilize at arity one, while every completely regular syntactic monoid stabilizes by arity two; finite group kernels show that the binary bound is sharp. At unary arity, every nonempty finite simple graph is realized by an explicit length-three language, yielding an exact chromatic-number formula and NP-completeness of deciding \(\operatorname{cmp}_1(L)\le 3\) for explicitly listed length-three languages. The structural boundary between compression heights one and two remains open.
Comments43 pages, 2 tables
