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.