# What is known about orbifolding ordered groups and sets? Who has been involved? Links to Lee metrics?

In mathematical music theory several ordered groups are considered. Some examples contain the frequency space or Tonnetzes. Other groups (commutative and non-commutative ones) are discussed by Dawid Lewin (Generalized Interval Systems and Transformations), Rudolf Wille (discussed also by Jan Thomas Winkler: Algebraische Modellierung von Tonsystemen), Wilfried Neumaier (Was ist ein Tonsystem?) or Marek Žabka.

Musicians are usually trained to distinguish complementary intervals such as fourth and fifth, music theorists usually treat them as the same. I'd like to investigate how the different points of view interact with each other. Mathematically this can be modelled as follows:

We consider a po-group $(\mathbb G,\leq)$, its linear order extensions and the quotient group $\mathbb G/N$ for a given normal subgroup $N\triangleleft \mathbb G$. As the musical application doesn't honour the order relation it's not sufficient to consider convex normal subgroups of $(\mathbb G,\leq)$. Another approach relational orbifolds has been discussed by Monika Zickwolf (Contr. Gen. Alg. 7) and Daniel Borchmann & Bernhard Ganter. Unfortunately both articles are interested in finite ordered sets and omit the case that is interesting for my work.

To illustrate the problem consider the set of integers $(\mathbb Z,\leq)$. A simple way of orbifolding leads to the binary relation $\{([a],[b]) \mid a,b \in \mathbb Z, a\leq b\} = \mathbb Z_{12}\times\mathbb Z_{12}$ (written as set of ordered pairs). On the other hand the same construction using the predecessor relation $\prec$ instead of the partial order $\leq$ leads to the binary relation $\{([a],[b])\mid a,b\in \mathbb Z, a= b-1\} = \{(x,y) \mid x,y\in\mathbb Z_{12}, y = x+1\}$, which again is a predecessor relation. But such a relation can not always be deduced.

As this example suggests the problem seems to be related to cyclic orderings, though the result might not necessarily be a cyclically ordered set. A counterexample can be constructed using the group $\mathbb G=\mathbb Z\times \mathbb Z$ and $N= \langle (0,4), (5,0)\rangle$ as the generated subgroup.

Which descriptions of such orbifolds exist in the language of relational structures?

I'm shure there exist descriptions both for automorphisms of ordered sets as well as using properties of ordered groups. But so far I'm lacking an elegant language.

Considering ordered groups one can try to separate in certain cases in $\mathbb G/N$ both a positive and a negative part, as well as some “intermixed” ones using nicely chosen transversal.

How can such a transversal be described?

Using these parts it seems to be possible to define a gerneralized Lee metric (C. Lee; IEEE Trans. Inform. Theory Vol. 4, pp. 77–82, Jun 1958).

Has someone discussed such metrics?

Edit: The correct definitions of the used term “orbifold” can be found in the paper by Borchmann and Ganter. I'll try to summarize it in short:

Here, an orbifold of a relational structure $(X,R)$ is a labelled relational structure $(X',R',\Lambda)$ such that $X'$ is a transversal of orbits of $X$ under the action of a subgroup $U$ of the automorphism group of $(X,R)$. $R'$ is a set of relations representing the relations of $R$ on $X'$ such that each tuple $(x_1,\ldots ,x_n)$ of a relation in R is mapped to the tuple $(x'_1,\ldots,x'_n)$ of the corresonding representatives in $X'$. The labelling $\Lambda$ assigns a labelling function to each of the relations, which maps each tuple to the set of automorphisms that must be used to reconstruct $(X,R)$ from $(X',R',\Lambda)$. Such a reconstruction is possible up to isomorphisms. Unfortunately folding ordered groups leads to many redundant loops and edges in the resultant graph, which blurr the interpretation in my case. In case of a po-group considering $(X',\leq')$ as a simple directed graph leads to the full graph on $X'$.

Edit2: Correctly the orbifold is defined as a tuple $\bigl(X',R',\Lambda, (U_t)_{t∈X'},U\bigr)$ where $U$ is the subgroup and $U_t$ are stabilizers of the elements of the transversal. In case of a reflexive binary relation $Λ(t,t)=U_t$. As the reflexive pairs are encoded in the relation $R'$, the annotation function can generally be abused to store the stabilizers of the set elements (that's why I used the short form above). As discussed in the comments the main purpose of that mapping is to store a local view on the group action.

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Added the music-theory tag. –  David Roberts Sep 14 '11 at 0:04
I think the first sentence should begin 'In mathematical music theory...' As it currently reads, it's hard not to respond 'Well, yes.' –  HJRW Sep 14 '11 at 9:24
Could you define what you mean by 'orbifold' in this context? I don't see the relation with the definition used in topology. –  HJRW Sep 14 '11 at 9:27
Thanks for the suggestions. I don't know why “music” had dissapeared. Orbifolding means folding along an orbit. I've tried to summarize the ideas of the mentioned literature. I didn't invent this name and I'm not familiar enough with the topological point of view to explain the relationship between the two notions. –  Tobias Schlemmer Sep 14 '11 at 16:31
Does $\Lambda$ by any chance assign to each tuple its stabiliser in $U$? –  HJRW Sep 15 '11 at 8:40