Categorical forms of the rigid direct-system representation for finite local-unit-aligned totally ordered monoids

RESUMO

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 $\tau$ induces a canonical $\tau$-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 $\tau$-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 $\tau$-compatible homomorphisms.