AI 中文总结
研究半格序半群上闭包算子相关的空间及诱导运算，证明\(X\)是谱空间等条件，给出诱导运算\(\star\)谱性的等价刻画与充分条件，还引入相关闭包算子并研究其性质及应用。
AI 中文摘要
设\(S\)为半格序半群，\(\mathrm{cl}\)为\(S\)上的闭包算子。考虑\(S\)的所有\(\mathrm{cl}\)-闭子集构成的空间\(X := \{A \in \mathcal{P}(S) \mid A^\mathrm{cl} = A\}\)，赋予由\(\mathcal{P}(S)\)上的壳核拓扑诱导的子空间拓扑。证明了\(X\)是谱空间且是\(\mathcal{P}(S)\)的逆紧子集当且仅当\(\mathrm{cl}\)是代数的。假设\(\mathrm{cl}\)是代数的，研究了由\(S\)上的乘法在\(X\)上诱导的运算\(A \star B := (AB)^\mathrm{cl}\)。主要结果给出了映射\(\star : X \times X \to X\)谱性的几个等价刻画。此外，从有限生成\(\mathrm{cl}\)-闭子集的良拟序条件得到\(\star\)谱性的一个充分条件。最后，引入了与半格序半群自然相关的三个闭包算子，研究了它们的代数和乘法性质，并将一般结果应用于相应空间和诱导运算。
英文摘要
Let $S$ be a semilattice-ordered semigroup and let $\mathrm{cl}$ be a closure operator on $S$. We consider the space $$X := \{A \in \mathcal{P}(S) \mid A^\mathrm{cl} = A\}$$ of all $\mathrm{cl}$-closed subsets of $S$, endowed with the subspace topology induced by the hull-kernel topology on $\mathcal{P}(S)$. We prove that $X$ is a spectral space and a retrocompact subset of $\mathcal{P}(S)$ if and only if $\mathrm{cl}$ is algebraic. Assuming that $\mathrm{cl}$ is algebraic, we then investigate the operation $$A \star B := (AB)^\mathrm{cl}$$ induced on $X$ by the multiplication on $S$. We obtain finite characterizations of spectrality of the map $\star: X \times X \to X$ in terms of finite subsets of $S$ and finitely generated $\mathrm{cl}$-closed subsets. Analogous criteria are established for left and right translations. We prove that if the underlying poset $(S, \leqslant)$ is well-quasi-ordered and the closure operator $\mathrm{cl}$ is algebraic and order-compatible, then the induced operation $\star$ is spectral. For the identity closure operator, we characterize spectrality of $\star$ by the finite decomposition property. We also show that spectrality of $\star$ implies spectrality of all left and right translations, whereas the converse fails in general, even for an algebraic multiplicative closure operator. Finally, we introduce three closure operators naturally associated with semilattice-ordered semigroups, study their algebraic and multiplicative properties, and apply the general results to the corresponding spectral spaces and induced operations.
Comments28 pages; substantially revised version. Corrected the statement concerning the relation between joint and separate spectrality. Added results on the identity closure operator and the finite decomposition property. Proofs and references have also been revised
