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

![enter image description here][1]

You can compose two such elements in two obvious ways:

![enter image description here][2]

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.

![enter image description here][3]

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 elements:

![enter image description here][4]

![enter image description here][5]

But when two sides of the same kind are distinguished:

![enter image description here][6]

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

![enter image description here][7]

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. 

![enter image description here][8]

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?


  [1]: https://i.sstatic.net/yPz1j.png
  [2]: https://i.sstatic.net/UqDK5.png
  [3]: https://i.sstatic.net/CVBmb.png
  [4]: https://i.sstatic.net/qQKZ5.png
  [5]: https://i.sstatic.net/oJ8cS.png
  [6]: https://i.sstatic.net/uYyp6.png
  [7]: https://i.sstatic.net/BGkxc.png
  [8]: https://i.sstatic.net/mZkeE.png