Canonical resolved-transport decomposition and reconstruction of local-unit-aligned ordered semigroups
Abstract
This paper develops a canonical decomposition and reconstruction theory for local-unit-aligned ordered semigroups. Two coherentizations of the positive-idempotent skeleton are introduced. The finer one records the closure forced within local-unit blocks, while the multiplication-coherent quotient yields a join-semilattice of canonical blocks. Components are the fibers of these blocks. For comparable blocks A≤ B, each positive idempotent q∈ B defines a transport homomorphism x↦ xq from the component over A to that over B. Keeping all such maps gives a resolved-transport family, which replaces the single connecting map used in an ordinary direct system. These canonical data reconstruct multiplication without additional assumptions: two elements are transported to their join component, and their product is the least product of corresponding transported images. They also determine every comparison directed from a lower component to a higher one. The full ambient order is recovered under any of three explicit order-recovery conditions. In particular, every totally ordered local-unit-aligned semigroup is completely reconstructed by its resolved-transport data. A complementary result recovers the order from componentwise order duality when a suitable component-preserving anti-automorphism is available. When each component receiving a proper transition has a least positive idempotent, the resolved family collapses to a single least-target map. Under a natural monotonicity condition, these maps form an ordinary direct system and induce a directed lexicographic order. Examples show both why the additional order conditions are needed and why resolved transports cannot in general be replaced by ordinary transition maps.
Related
- On the Jacobs-de Leeuw-Glicksberg decomposition for representations of semigroups
- Heteroskedastic Canonical Polyadic Tensor Decomposition
- Log Canonical Counterexamples to the Chen--Jiang Decomposition
- On the Dilation Theory and Canonical Decomposition of Θn-Contractions
- Particle-Number-Preserving Quantum Signal Processing for Thermal Properties: Canonical-Ensemble Estimation and Grand-Canonical Reconstruction
BibTeX
@misc{jenei2026canonical,
title = {{Canonical resolved-transport decomposition and reconstruction of local-unit-aligned ordered semigroups}},
author = {Sándor Jenei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20255},
eprint = {2609.20255},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20255},
}
RIS
TY - UNPB
TI - Canonical resolved-transport decomposition and reconstruction of local-unit-aligned ordered semigroups
AU - Sándor Jenei
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.20255
UR - https://arxiv.org/abs/2609.20255
AB - This paper develops a canonical decomposition and reconstruction theory for local-unit-aligned ordered semigroups. Two coherentizations of the positive-idempotent skeleton are introduced. The finer one records the closure forced within local-unit blocks, while the multiplication-coherent quotient yields a join-semilattice of canonical blocks. Components are the fibers of these blocks. For comparable blocks A≤ B, each positive idempotent q∈ B defines a transport homomorphism x↦ xq from the component over A to that over B. Keeping all such maps gives a resolved-transport family, which replaces the single connecting map used in an ordinary direct system. These canonical data reconstruct multiplication without additional assumptions: two elements are transported to their join component, and their product is the least product of corresponding transported images. They also determine every comparison directed from a lower component to a higher one. The full ambient order is recovered under any of three explicit order-recovery conditions. In particular, every totally ordered local-unit-aligned semigroup is completely reconstructed by its resolved-transport data. A complementary result recovers the order from componentwise order duality when a suitable component-preserving anti-automorphism is available. When each component receiving a proper transition has a least positive idempotent, the resolved family collapses to a single least-target map. Under a natural monotonicity condition, these maps form an ordinary direct system and induce a directed lexicographic order. Examples show both why the additional order conditions are needed and why resolved transports cannot in general be replaced by ordinary transition maps.
ER -