AI 中文总结
本文研究有限局部单位对齐全序幺半群的刚性直接系统表示的范畴性，证明其对严格块态射和固有$\ au$-相容同态均为自函子，并给出相应等价性。
AI 中文摘要
我们研究了有限局部单位对齐全序幺半群的典范刚性直接系统表示的自函子与范畴结构。局部单位映射$\ au$诱导一个典范的$\ au$-乘法协调分解为分量幺半群，且相关表示通过一个有限链索引的刚性直接系统重构原始有序幺半群，其真转移映射为单位常数。本文确定了该表示为范畴性的态射类。首先，我们证明了具有严格块态射的有限局部单位对齐全序幺半群与具有有向序相容系统态射的刚性直接系统之间的等价性。其次，我们证明了$\ au$-相容同态的固有等价性，即与局部单位映射交换的保序单位同态。在第二种情形中，不同的正幂等元以及因此不同的典范分量可能坍缩到单一目标分量；在直接系统侧，这由非单射的保序指标映射连同满足相应坍缩与吸收公理的分量映射来表示。因此，典范刚性直接系统表示对于严格块态射和固有$\ au$-相容同态都是自函子的。
英文摘要
We study the functorial and categorical structure of the canonical rigid direct-system representation of finite local-unit-aligned totally ordered monoids. The local-unit map $τ$ induces a canonical $τ$-multiplication-coherent decomposition into component monoids, and the associated representation reconstructs the original ordered monoid from a finite chain-indexed rigid direct system whose proper transition maps are unit-constant. The present paper identifies the morphism classes for which this representation is categorical. First, we prove an equivalence between finite local-unit-aligned totally ordered monoids with strict block morphisms and rigid direct systems with directed-order-compatible system morphisms. Second, we prove an intrinsic equivalence for $τ$-compatible homomorphisms, that is, isotone unital homomorphisms commuting with the local-unit map. In this second setting, distinct positive idempotents and hence distinct canonical components may collapse to a single target component; on the direct-system side this is represented by non-injective isotone index maps together with component maps satisfying the corresponding collapse and absorption axioms. Thus the canonical rigid direct-system representation is functorial both for strict block morphisms and for intrinsic $τ$-compatible homomorphisms.
