I intended to call this series of posts “What group is this?”, but I switched to this more precise, albeit more bland name. In this first post of the series I take again, in more generality, the construction explained in the post Towards geometric Plünnecke graphs.
The construction starts in the same way, almost. After I give this first part of the construction, an interpretation in term sof groupoids is provided. We consider only the moves R1a and R2a, like in the post “A roadmap to computing with space“:
(The names “R1a”, “R2a” come from the names of oriented Reidemeister moves, see arXiv:0908.3127 by M. Polyak.)
Definition 1. The moves R1a, R2a act on the set of binary trees with nodes decorated with two colours (black and white) and leaves decorated with elements of a set of “variable names” which has at least two elements. I shall denote by … such trees and by … elements of .
The edges of the trees are oriented upward. We admit to be a subset of , thinking about as an edge pointing upwards which is also a leaf decorated with .
The moves are local, i.e. they can be used for any portion of a tree from which looks like one of the patterns from the moves, with the understanding that the rest of the respective tree is left unchanged.
We denote by the fact that the can be transformed into by a finite sequence of moves.
Definition 2. The class of finite trees is the smallest subset of with the properties:
- if then , where is the tree
- if then and , where is the tree
and is the tree
- if and we can pass from to (i.e. ) by one of the moves then .
Definition 3. Two graphs are close, denoted by , if there is such that can be moved into .
Notice that then .
Proposition 1. The closeness relation is an equivalence.
Proof. I start with the remark that if and only if , where is the tree
Indeed, if there is such that . Then
which proves that . Then for any , because , therefore . Suppose now that . Then . Notice that , by the following sequence of moves:
But , from the hypothesis. Therefore , which is equivalent with .
Finally, suppose that , . Then by the previous reasoning. Then there are such that and . It follows that , therefore , which proves that .
Definition 4. The class of finite points of is is the set of equivalence classes w.r.t .
Same construction, with groupoids. We may see as being an equivalence relation. Let be the set of equivalence classes w.r.t . We can define on the operations and (because the moves R1a, R2a are local). Then is the free left idempotent right quasigroup generated by the set .
Idempotent right quasigroups are the focus of the article arXiv:0907.1520, where emergent algebras are introduced as deformations of such objects. An idempotent right quasigroup is a non-empty set endowed with two operations, such that
- (idempotence) for any ,
- (right quasigroup) for any
Let be the trivial (pair) groupoid over . This is the groupoid with objects which are elements of and arrows of the form . Equivalently, we see to be the set of it’s arrows, we identify objects with their identity arrows (in this case we identify with it’s identity arrow ). Seen like this, the trivial groupoid is just the set , with the partially defined operation (composition of arrows)
and with the unary inverse operation
Remark that the function defined by is a bijection of the set of arrows and moreover
- it preserves the objects ,
- the inverse has the expression .
Define the groupoid by declaring to be an isomorphism of groupoids. This means to be the set of arrows , with the partially defined composition of arrows given by
for any pair of arrows such that can be composed in with , and unary inverse operation given by
The groupoid has then the composition operation
the unary inverse operation
and the set of objects .
Consider the set , seen as a subset of arrows of the groupoid .
The class of finite trees appears in the following way. First define to be the set of equivalence classes w.r.t of elements in .
Remark that is a sub-groupoid of , which moreover it contains and is closed w.r.t. the application of , seen this time as a function (which is not a morphism) from to itself. In fact is the smallest subset of with this property. Let’s give to the groupoid the name , seen as a sub-groupoid of .
Moreover is a sub-groupoid of the trivial groupoid , with set of objects . But sub-groupoids of the trivial groupoid are the same thing as equivalence relations. In this particular case if and only if and .
Next time you’ll see some groups (which are associated to parallel transport in dilation structures) which are in some sense universal, but I don’t know (yet) what structure they have. “What group is this?” I shall ask next time.
Do you remark at which stage of this construction the map becomes the territory, thus creating points out of abstract nonsense?
To get a sense of this, replace the set of arrows with a graph with nodes in .