Contexts and notations for composing asymmetric simplices

Question

Imagine the elements of a group-like structure as puzzle pieces with essential two sides, an IN-side and an OUT-side.

You can compose two such pieces in two obvious ways:

Now consider triangular puzzle pieces with at least one IN- and one OUT-side. These are 2-simplices with a non-trivial partition of their sides.

As long as two sides of the same kind are not distinguished (i.e. the simplices are symmetric), there are again two ways to compose two such pieces:

But when two sides of the same kind are distinguished:

a single operator + doesn't suffice anymore. One has to specify which of the (eventually) two OUT-sides of the first piece is to be plugged into which of the (eventually) two IN-sides of the second piece:

I wonder:

(1) In which specific (algebraic or simplicial resp. topological) contexts do such asymmetric pieces appear?

(2) How then is the problem of notation solved, especially: how are "words" (conglomerates) of such pieces symbolically written down (which is trivial for group-like structures and symmetric simplices by the use of + or $\circ$ or even no symbol at all).

Note that the composition is supposed to be in a natural way associative.

A related question concerns the possibility that cycles are allowed.

For group-like structures, cycles are not allowed (and in the rigid picture of puzzle pieces cycles are not even possible), for simplex-based structures cycles are supposed to be allowed:

(3) How is the problem of notation solved for possibly circular conglomerates?

Answer 1

Here is an attempt. Not a full answer, rather a suggestion of an approach.

Instead of triangles let us look at nodes with three strings coming out of them. So, for the triangle of type "A" let us write

and for the triangle of type "B" let us write

The bottom line is that the strings that go upwards from the node correspond to OUT-sides, and strings that go downwards from the node correspond to IN-sides. Let us regard these pictures as string diagrams http://ncatlab.org/nlab/show/string+diagram. Then we can compose them as we compose string diagrams. Some of the possible compositions are:

We allow strings to intersect each other (as is the case in the leftmost diagram). Such compositions should correspond to the "triangle compositions". So as the leftmost diagram above is "A1 + 0B", the second diagram is "A1 + 1B" and the third diagram is "B + A" as in OP.

Triangle tilings which are obtained inductively by adjoining one tile by a single edge at a time are representable by string diagrams. On the other hand, a tiling can be constructed from a string diagram that has no cycles. These can be shown by induction on number of tiles/nodes. I believe that there is a one-to-one correspondence between tilings and string diagrams of these types.

For more complicated tilings things get more difficult. Anyway, the "cycle" in the OP would be

After this category theory and string diagrams tell us how to solve the problem of notation. We can imagine that we are inside a symmetric monoidal category where we have an object $X$ and two morphisms $f : X\otimes X \rightarrow X$ and $g : X \rightarrow X\otimes X$ corresponding to the triangles "A" and "B" respectively (more concretely say that we are in the symmetric monoidal category free on such a data). Then the notation comes from the string calculus for a symmetric monoidal category. Thus for example, "A1 + 0B" would be the composite

$$X\otimes X \xrightarrow{1_X\otimes g} X\otimes X\otimes X \xrightarrow{s\otimes 1_X} X\otimes X\otimes X \xrightarrow{1_X\otimes f} X\otimes X,$$

where $s$ stands for the symmetry of the monoidal category.