A canonical rigid direct-system representation of finite local-unit-aligned totally ordered monoids

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 τ , through the canonical stratification of the positive idempotent skeleton into τ τ -multiplication-coherent blocks. More precisely, from τ 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 τ -multiplication-cohesive and yields a Clifford-type ordinal-sum-like reconstruction theorem for finite local-unit-aligned totally ordered monoids.