AI 中文总结
本文在有界格上为四类一致模建立了充分必要的表示定理，通过中性元分解与内部/闭包算子等工具刻画其结构，并比较了各类间关系及有界格特化。
AI 中文摘要
本文在有界格上为一致模类 $\widetilde{\mathcal{U}}_{\mathrm{top}}$、$\widetilde{\mathcal{U}}_{\mathrm{bot}}$、$\widetilde{\mathcal{U}}_{\mathrm{max}}^{r}$ 和 $\widetilde{\mathcal{U}}_{\mathrm{min}}^{r}$ 建立了充分必要的表示定理，将相应的有界格结果进行了推广。对于不属于任何非平凡循环的非极值中性元 $e$，我们使用与 $e$ 不可比较元素的公共分解；证明了每个一致模的中性元都是中间传递的。$\widetilde{\mathcal{U}}_{\mathrm{top}}$（相应地，$\widetilde{\mathcal{U}}_{\mathrm{bot}}$）中的每个一致模可由一个内部算子（相应地，一个闭包算子）、一个分量一致模和一个递增、结合且交换的运算表示，且所有分量都是唯一确定的。另外两个条件刻画了子类 $\widetilde{\mathcal{U}}_{\mathrm{top}}^{\star}$ 和 $\widetilde{\mathcal{U}}_{\mathrm{bot}}^{\star}$。对于 $\widetilde{\mathcal{U}}_{\mathrm{max}}^{r}$（相应地，$\widetilde{\mathcal{U}}_{\mathrm{min}}^{r}$），我们确定了 $I_e^3\times I_e^3$ 中取值位于 $[0,e[\cup I_e^1$（相应地，$]e,1]\cup I_e^2$）或等于 $e$ 的序对。产生 $e$ 的序对满足对称的唯一伙伴条件，而其余值由部分运算编码。区域序相容性和显式结合性条件随后给出了充分必要的表示。最后，我们比较了这四个类，将它们与相应的有界偏序集类联系起来，证明了它们在传递性下的有界格特化，并提供了真格上的有限例子以说明额外的相容性条件。
英文摘要
In this paper, we establish necessary and sufficient representation theorems for the classes $\widetilde{\mathcal{U}}_{\mathrm{top}}$, $\widetilde{\mathcal{U}}_{\mathrm{bot}}$, $\widetilde{\mathcal{U}}_{\mathrm{max}}^{r}$, and $\widetilde{\mathcal{U}}_{\mathrm{min}}^{r}$ of uninorms on bounded trellises, extending the corresponding bounded-lattice results. For a non-extremal neutral element $e$ belonging to no non-trivial cycle, we use a common decomposition of the elements incomparable with $e$; every neutral element of a uninorm is shown to be middle-transitive. Each uninorm in $\widetilde{\mathcal{U}}_{\mathrm{top}}$ (respectively, $\widetilde{\mathcal{U}}_{\mathrm{bot}}$) is represented by an interior operator (respectively, a closure operator), a component uninorm, and an increasing, associative, and commutative operation, with all components uniquely determined. Two further conditions characterize the subclasses $\widetilde{\mathcal{U}}_{\mathrm{top}}^{\star}$ and $\widetilde{\mathcal{U}}_{\mathrm{bot}}^{\star}$. For $\widetilde{\mathcal{U}}_{\mathrm{max}}^{r}$ (respectively, $\widetilde{\mathcal{U}}_{\mathrm{min}}^{r}$), we identify the pairs in $I_e^3\times I_e^3$ whose values lie in $[0,e[\cup I_e^1$ (respectively, $]e,1]\cup I_e^2$) or equal $e$. The pairs producing $e$ satisfy a symmetric unique-partner condition, while the remaining values are encoded by a partial operation. Regional order compatibility and explicit associativity conditions then yield necessary and sufficient representations. Finally, we compare the four classes, relate them to the corresponding bounded-psoset classes, prove their bounded-lattice specialization under transitivity, and provide finite examples on proper trellises illustrating the additional compatibility conditions.
