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arXiv · 2607.23801

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

Sándor Jenei
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Abstract

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

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Sándor Jenei. 2026-07-26. A canonical rigid direct-system representation of finite local-unit-aligned totally ordered monoids. https://arxiv.org/abs/2607.23801

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