Emergent algebras as combinatory logic (Part I)
At some point this new thread I am starting now will meet the Towards qubits thread.
Definition 1. Let be a commutative group with neutral element denoted by and operation denoted multiplicatively. A idempotent quasigroup is a set endowed with a family of operations , indexed by , such that:
- For any the pair is an idempotent quasigroup,
- The operation is trivial: for any we have ,
- For any and any we have: .
This definition may look strange, let me give some examples of idempotent quasigroups.
Example 1. Real vector spaces: let be a real vector space, with multiplication of reals as operation. We define, for any the following quasigroup operation:
These operations give to the structure of a $(0,+\infty)$ idempotent quasigroup. Notice that is the dilation based at , of coefficient , applied to .
Example 2. Complex vector spaces: if is a complex vector space then we may take and we continue as previously, obtaining an example of a idempotent quasigroup.
Example 3. Contractible groups: let be a group endowed with a group morphism . Let with the operation of addition of integers (thus we may adapt Definition 1 to this example by using “” instead of “” and “” instead of “” as the name of the neutral element of ). For any let
This a idempotent quasigroup. The most interesting case (relevant also for Definition 3 below) is the one when is an uniformly contractive automorphism of the topological group . The structure of these groups is an active exploration area, see for example arXiv:0704.3737 by Helge Glockner and the bibliography therein (a fundamental result here is Siebert article Contractive automorphisms on locally compact groups, Mathematische Zeitschrift 1986, Volume 191, Issue 1, pp 73-90). See also conical groups and relations between contractive and conical groups introduced in arXiv:0804.0135, shortly explained in arXiv:1005.5031.
Example 5. A group with an invertible self-mapping such that , where is the identity of the group . In this case the construction from Example 3 works here as well because there is no need for to be a group morphism.
Example 6. Local versions. We may accept that there is a way (definitely needing care to well formulate, but intuitively cleart) to define a local version of the notion of a idempotent quasigroup. With such a definition, for example, a convex subset of a real vector space gives a local $(0,+\infty)$ idempotent quasigroup (as in Example 1) and a neighbourhood of the identity of a topological group , with an identity preserving, locally defined invertible self map (as in Example 5) gives a local idempotent quasigroup.
Example 7. A particular case of Example 6, is a Lie group with the operations defined for any by
Example 8. A less symmetric example is the one of being a riemannian manifold, with associated operations defined for any by
Example 9. More generally, any metric space with dilations (introduced in arXiv:math/0608536[MG] ) is a local idempotent quasigroup.
Example 10. One parameter deformations of quandles. A quandle is a self-distributive quasigroup. Take now a one-parameter family of quandles (indexed by ) which satisfies moreover points 2. and 3. from Definition 1. What is interesting about this example is that quandles appear as decorations of knot diagrams, which are preserved by the Reidemeister moves. At closer examination, examples 1-4 are all particular cases of one parameter quandle deformations!
I shall define now the operations of approximate sum and approximate difference associated to a idempotent quasigroup.
For any , let use define .
Definition 2. The approximate sum operation is (for any )
The approximate difference operation is (for any )
Suppose now that is a separable uniform space. Let us suppose that the commutative group $\Gamma$ is a topological group endowed with an absolute, i.e. with an invariant topological filter, denoted by . We write for a net in $\latex \Gamma$ which converges to the filter . The image to have in mind is with multiplication of reals as operation and with the filter as the filter generated by sets with . This filter is the restriction to the set of the filter of neighbourhoods of the number . Another example is with addition of integers as operation, seen as a discrete topological group, with the absolute generated by sets for all . For this example the neutral element (denoted by in Definition 1) is the integer , therefore in this case we can change notations from multiplication to addition, becomes , the absolute becomes , and so on.
Definition 3. An emergent algebra (or uniform idempotent quasigroup) is a idempotent quasigroup , as in Definition 1, which satisfies the following topological conditions:
- The family of operations is compactly contractive, i.e. for any compact set , for any and for any open neighbourhood of , there is an open set which belongs to the absolute such that for any and we have .
- As there exist the limits
and moreover these limits are uniform with respect to in compact sets.
The structure theorem of emergent algebras is the following:
Theorem 1. Let be a emergent algebra. Then for any the pair is a conical group.
In the next post on this subject I shall explain why this is true, in the language of graphic lambda calculus.