  !-graphs provide a means of reasoning about infinite families of string
diagrams and have proven useful in manipulation of (co)algebraic structures
like Hopf algebras, Frobenius algebras, and compositions thereof. However, they
have previously been limited by an inability to express families of diagrams
involving non-commutative structures which play a central role in algebraic
quantum information and the theory of quantum groups. In this paper, we fix
this shortcoming by offering a new semantics for non-commutative !-graphs using
an enriched version of Penrose's abstract tensor notation.
