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 multiplication, respectively. This led to new characterization of 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 homomorphisms. Constructing classifying invariants $\mathsf{Graph} which are homomorphisms, additive join, multiplicative is therefore major interest. We call such semiring-homomorphic. The only known fractional nature: chromatic number, projective rank, Haemers bounds, well Lov\'asz number (with latter two evaluated on complementary graph). Here, we provide unified construction these based linear-like families graphs. Along way, also investigate additional algebraic structure corresponding fractionalization. Linear-like concept combinatorial geometry different from matroids may be independent