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 two kinds of sides - IN and OUT - 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 A are two be plugged into which of the (eventually) two IN-sides of B:

![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" of such pieces symbolically written down (which is trivial for group-like structures 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**.


  [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