I am going over the publicly available pdf of "Div, Grad, curl are dead", in chapter-2 , section-11, the algebra of one forms is introduced (see page-37). In it, I understood how to take negative of a one form, we just put a tick on the line, and also the scaling features (put the two lines closer) but I can't get how the addition rule is supposed to be.
The author has given the following picture on the page:
Hopefully someone can give a more elaborate explanation of the icon addition scheme shown above
For an elementary understanding, translate it into vectors, by naming $A,B,C$ the vertices of the triangle:
The first one is
$$\vec{AC}=\vec{AB}+\vec{BC} \iff C-A=(B-A)+(C-B)$$
The second one is
$$\vec{AB}+\vec{BC}+\vec{CA}=0 \iff (B-A)+(C-B)+(A-C)=0$$
which are clearly equivalent.
Remark: the second notation in (1) and (2) comes from a more abstract point of view, necessitating to consider homology chains, with boundary operator $\partial$ (see here.