AI 中文总结
研究有限局部单位对齐全序幺半群，通过局部单位映射\(\tau\)构建严格兼容的有限链索引直系统，证明其有典范刚性直系统表示且能重构该幺半群，还得出有限情况下的相关定理。
AI 中文摘要
我们研究有限局部单位对齐全序幺半群，即每个元素的最大右局部单位和左局部单位一致的有限全序幺半群。我们证明每个这样的幺半群都有一个典范刚性链索引直系统表示，反之，每个相应类型的刚性系统都能重构一个有限局部单位对齐全序幺半群。该表示由局部单位映射\(\tau\)通过正幂等骨架到\(\tau\)乘法相干块的典范分层内在诱导。具体而言，从\(\tau\)构建组成幺半群和过渡映射，形成一个严格兼容的有限链索引直系统，由此恢复环境序和环境乘法。在有限情况下，严格兼容性迫使每个恰当过渡映射为单位常数；这种刚性使典范分量为\(\tau\)乘法凝聚，并为有限局部单位对齐全序幺半群产生一个类似 Clifford 型序数和的重构定理。
英文摘要
We study finite local-unit-aligned totally ordered monoids, that is, finite totally ordered monoids in which each element has coinciding greatest right and left local units. We prove that every such monoid admits a canonical rigid chain-indexed direct-system representation, and conversely that every rigid system of the corresponding kind reconstructs a finite local-unit-aligned totally ordered monoid. The representation is induced intrinsically by the local-unit map $τ$, through the canonical stratification of the positive idempotent skeleton into $τ$-multiplication-coherent blocks. More precisely, from $τ$ we construct component monoids and transition maps forming a strictly compatible finite chain-indexed direct system from which both the ambient order and the ambient multiplication are recovered. In the finite case, strict compatibility forces every proper transition map to be unit-constant; this rigidity makes the canonical components $τ$-multiplication-cohesive and yields a Clifford-type ordinal-sum-like reconstruction theorem for finite local-unit-aligned totally ordered monoids.
