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 pieces 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 pieces: ![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 piece is to be plugged into which of the (eventually) two IN-sides of the second piece: ![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