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

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You can compose two such pieces in two obvious ways:

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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.

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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

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But when two sides of the same kind are distinguished:

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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

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.

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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?

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  • $\begingroup$ I do not understand why you identify the two ways of sticking symmetric triangles together in your fourth picture. Both give rise to different arrangements if you add further triangles. $\endgroup$ Oct 17, 2015 at 8:04
  • $\begingroup$ True enough! But regarding piece B as a symmetric one you cannot distinguish the two arrangements - with respect to B. That's what I wanted to point out. $\endgroup$ Oct 19, 2015 at 23:11

1 Answer 1

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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

enter image description here

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:

enter image description here

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

enter image description here

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.

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  • $\begingroup$ Does every such diagram yield a legal two-dimensional arrangement of tiles? If not, can you see from the diagrams if they can be realised in 2d without actually constructing realisations? $\endgroup$ Oct 22, 2015 at 6:33
  • $\begingroup$ No they don't. The answer is perhaps only a suggestion of an approach. In fact, I don't know if more complicated planar diagrams can be represented by diagrams. $\endgroup$ Oct 22, 2015 at 12:21
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    $\begingroup$ @Sebastian See my last edit too. $\endgroup$ Oct 22, 2015 at 15:41
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    $\begingroup$ @Dimitri: Thanks, sounds promising. $\endgroup$ Oct 23, 2015 at 10:05

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