This note attempts to understand graph limits as defined by Lovasz and Szegedy in terms of harmonic analysis on semigroups. This is done by representing probability distributions of random exchangeable graphs as mixtures of characters on the semigroup of unlabeled graphs with node-disjoint union, thereby providing an alternative derivation of de Finetti's theorem for random exchangeable graphs.
We study numerical semigroups $S$ with the property that if $m$ is the multiplicity of $S$ and $w(i)$ is the least element of $S$ congruent with $i$ modulo $m$, then $0<w(1)<\dots <w(m-1)$. The set of numerical semigroups with this property and fixed multiplicity is bijective with an affine semigroup and consequently it can be described by a finite set of parameters. Invariants like the gender, type, embedding dimension and Frobenius number are computed for several families of this kind of numerical semigroups.