The post Axioms for projective conical spaces (towards qubits II) introduces a generalization of projective spaces to projective conical space. These are a kind of non-commutative version of projective spaces, exactly in the same sense as the one that affine conical spaces are a non-commutative generalization of affine spaces.
That post has been done
before the discovery of graphic lambda calculus. [UPDATE: no, I see that it was done after, but GLC was not used in that post.]
Now, the beautiful thing is that all the 4 axioms of projective conical spaces have the same form, if represented according to the same ideas as the ones of graphic lambda calculus.
There will be more about this, but I show you for the moment only how the first part of (PG1) looks like, in the original version and in the new version.
Here is the first part of (PG2) in old and new versions.