AI 中文总结
研究平衡剩余偏序半群，通过构造剩余相干化$\mathcal C_{\mathrm r}(\mathbf M)$，给出规范分解 - 重构理论，其块定义规范分量，商骨架记录目标分量，还与子半格稳定可见性分解比较，显示前者划分更精细。
AI 中文摘要
本文为平衡剩余偏序半群发展了一种规范的分解 - 重构理论。出发点是内在局部单位映射$\tau(x)=x\backslash x=x/x$，其值为正幂等元。该映射的原始纤维通常过细，与乘法和剩余不兼容。因此构造了剩余相干化$\mathcal C_{\mathrm r}(\mathbf M)$，它是正幂等骨架的最精细商，在商层面上这三个局部单位输出有良好定义。$\mathcal C_{\mathrm r}(\mathbf M)$的块定义了规范分量，商骨架记录了乘积和剩余的目标分量。这些数据与分量代数、乘积影子映射$x\mapsto xq$和剩余影子映射$y\mapsto y/q$一起重构原始代数。最后将此构造与子半格稳定可见性分解进行比较，在诱导迭代分解的每个有限阶段，由剩余相干化得到的划分比任何子半格稳定可见性选择得到的划分更精细。
英文摘要
This paper develops a canonical decomposition--reconstruction theory for balanced residuated partially ordered semigroups. The starting point is the intrinsic local-unit map $τ(x)=x\backslash x=x/x, $ whose values are positive idempotents. The primitive fibres of this map are generally too fine to be compatible with multiplication and residuals: the local units of $xy$, $x\backslash y$, and $x/y$ need not be determined by the local units of $x$ and $y$. We therefore construct the residual coherentization $\mathcal C_{\mathrm r}(\mathbf M)$, the finest quotient of the positive-idempotent skeleton on which these three local-unit outputs are well defined at quotient level. The blocks of $\mathcal C_{\mathrm r}(\mathbf M)$ define the canonical components, while the quotient skeleton records the target component for products and residuals. Together with the component algebras, the product-shadow maps $x\mapsto xq$, and the residual-shadow maps $y\mapsto y/q$, these data reconstruct the original algebra. The final part compares this construction with subsemilattice-steady visibility decompositions. At every finite stage of the induced iterative decompositions, the partition obtained from residual coherentization is finer than the partition obtained from any subsemilattice-steady visibility choice. Equivalently, each component produced by the subsemilattice-steady construction is a union of residual-coherent components.
