  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.
