A unified construction of semiring-homomorphic graph invariants

It has recently been observed by Zuiddam that finite graphs form a preordered commutative semiring under the graph homomorphism preorder together with join and disjunctive product as addition and multiplication, respectively. This led to a new characterization of the Shannon capacity $\Theta$ via Strassen's Positivstellensatz: $\Theta(\bar{G}) = \inf_f f(G)$, where $f : \mathsf{Graph} \to \mathbb{R}_+$ ranges over all monotone semiring homomorphisms. Constructing and classifying graph invariants $\mathsf{Graph} \to \mathbb{R}_+$ which are monotone under graph homomorphisms, additive under join, and multiplicative under disjunctive product is therefore of major interest. We call such invariants semiring-homomorphic. The only known such invariants are all of a fractional nature: the fractional chromatic number, the projective rank, the fractional Haemers bounds, as well as the Lov\'asz number (with the latter two evaluated on the complementary graph). Here, we provide a unified construction of these invariants based on linear-like semiring families of graphs. Along the way, we also investigate the additional algebraic structure on the semiring of graphs corresponding to fractionalization. Linear-like semiring families of graphs are a new concept of combinatorial geometry different from matroids which may be of independent interest.