Categorical forms of the rigid direct-system representation for finite local-unit-aligned totally ordered monoids
Abstract
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.
Related
- A Categorical Realization of the (2-)Category of Monoids via Sch{ü}tzenberger Categories and Strict Factorization Systems
- Canonical resolved-transport decomposition and reconstruction of local-unit-aligned ordered semigroups
- Higher representation stability for ordered configuration spaces
- Presentation of monoids generated by a projection and an involution
- Generation of Iterated Wreath Products Constructed from Full Transformation Monoids and Symmetric Groups
BibTeX
@misc{jenei2026categorical,
title = {{Categorical forms of the rigid direct-system representation for finite local-unit-aligned totally ordered monoids}},
author = {Sándor Jenei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20233},
eprint = {2609.20233},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20233},
}
RIS
TY - UNPB
TI - Categorical forms of the rigid direct-system representation for finite local-unit-aligned totally ordered monoids
AU - Sándor Jenei
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.20233
UR - https://arxiv.org/abs/2609.20233
AB - 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.
ER -