The semiring of spectral duality skeletons: cancellation, characters and factorization
Abstract
Weighted dual skeletons are the combinatorial core of spectral duality structures. Their isomorphism classes, under Cartesian product and disjoint union, form a semiring graded by the degree. We identify it completely: a class is determined by its degree, a symmetric element of the group semiring N_0[R] and its number of fixed points. Equivalently, skeletons of a given degree are finite C_2-sets over the C_2-monoid (R,+), these data are their marks, and the semiring is an infinite-monoid analogue of a crossed Burnside semiring. The multiplicative monoid is not cancellative, the only obstruction being that multiplication by a skeleton without fixed points forgets the fixed points; accordingly, the Grothendieck ring has explicit zero divisors. A continuous family of semiring characters, with the degree and the number of fixed points, is a complete invariant, and for commensurable ratios the multiplicative Grothendieck group is described by Pólya's theorem. The main subject is factorization. The reduced monoid is atomic with bounded lengths, but factorization is not unique, the elasticity is infinite, and even the divisor-closed submonoid with one fixed point is not half-factorial. Without fixed points, every reducible skeleton splits off a single orbit, which gives a criterion by centers, exactly when the number of orbits is 8, 16, a prime or twice a prime. Irreducibility is generic in almost every type. Progression skeletons factor according to mixed-radix number systems, with all factorizations of equal length. Finally, orbit counting gives the dimension of the separatrix and the response rank of products.
Related
- Spectral Equality for Tropical Totally Nonnegative Matrices: Dominant Cancellation and Newton Polygons
- Spectral Deflation for Factorization-Free Matrix Filtering in Muon and Semidefinite Programming
- Scalable Approximate Selected Inversion Based on Single-Level Incomplete LDLT Factorization and Spectral Corrections for Large Sparse Systems
- Spectral Gap of the Hexagonal AKLT Model via Boundary-State Factorization
- From Walks to Skeletons: A Reusable Transport Architecture for Interactive Monte Carlo PDE Solvers
BibTeX
@misc{coso2026semiring,
title = {{The semiring of spectral duality skeletons: cancellation, characters and factorization}},
author = {Juan Antonio Vega Coso},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23684},
eprint = {2610.23684},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23684},
}
RIS
TY - UNPB
TI - The semiring of spectral duality skeletons: cancellation, characters and factorization
AU - Juan Antonio Vega Coso
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23684
UR - https://arxiv.org/abs/2610.23684
AB - Weighted dual skeletons are the combinatorial core of spectral duality structures. Their isomorphism classes, under Cartesian product and disjoint union, form a semiring graded by the degree. We identify it completely: a class is determined by its degree, a symmetric element of the group semiring N_0[R] and its number of fixed points. Equivalently, skeletons of a given degree are finite C_2-sets over the C_2-monoid (R,+), these data are their marks, and the semiring is an infinite-monoid analogue of a crossed Burnside semiring. The multiplicative monoid is not cancellative, the only obstruction being that multiplication by a skeleton without fixed points forgets the fixed points; accordingly, the Grothendieck ring has explicit zero divisors. A continuous family of semiring characters, with the degree and the number of fixed points, is a complete invariant, and for commensurable ratios the multiplicative Grothendieck group is described by Pólya's theorem. The main subject is factorization. The reduced monoid is atomic with bounded lengths, but factorization is not unique, the elasticity is infinite, and even the divisor-closed submonoid with one fixed point is not half-factorial. Without fixed points, every reducible skeleton splits off a single orbit, which gives a criterion by centers, exactly when the number of orbits is 8, 16, a prime or twice a prime. Irreducibility is generic in almost every type. Progression skeletons factor according to mixed-radix number systems, with all factorizations of equal length. Finally, orbit counting gives the dimension of the separatrix and the response rank of products.
ER -