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 $\Gamma$  be a commutative group with neutral element denoted by $1$ and operation denoted multiplicatively. A $\Gamma$ idempotent quasigroup is a set $X$ endowed with a family of operations $\circ_{\varepsilon}: X \times X \rightarrow X$,  indexed by $\varepsilon \in \Gamma$, such that:

1. For any $\varepsilon \in \Gamma$ the pair $(X, \circ_{\varepsilon})$ is an idempotent quasigroup,
2. The operation $\circ_{1}$ is trivial: for any $x,y \in X$ we have $x \circ_{1} y = y$,
3. For any $x, y \in X$ and any $\varepsilon, \mu \in \Gamma$ we have:  $x \circ_{\varepsilon} ( x \circ_{\mu} y) = x \circ_{\varepsilon \mu} y$.

This definition may look strange, let me give some examples of $\Gamma$ idempotent quasigroups.

Example 1.  Real vector spaces: let $X$ be a real vector space, $\Gamma = (0,+\infty)$ with multiplication of reals as operation. We define, for any $\varepsilon > 0$ the following quasigroup operation:

$x \circ_{\varepsilon} y = (1-\varepsilon) x + \varepsilon y$

These operations give to $X$ the structure of a $(0,+\infty)$ idempotent quasigroup.  Notice that $x \circ_{\varepsilon}y$ is the dilation based at $x$, of coefficient $\varepsilon$, applied to $y$.

Example 2. Complex vector spaces: if $X$ is a complex vector space then we may take $\Gamma = \mathbb{C}^{*}$ and we continue as previously, obtaining an example of a $\mathbb{C}^{*}$ idempotent quasigroup.

Example 3.  Contractible groups: let $G$ be a group endowed with a group morphism $\phi: G \rightarrow G$. Let $\Gamma = \mathbb{Z}$ with the operation of addition of integers (thus we may adapt Definition 1 to this example by using “$\varepsilon + \mu$” instead of “$\varepsilon \mu$” and “$0$” instead of “$1$” as the name of the neutral element of $\Gamma$).  For any $\varepsilon \in \mathbb{Z}$ let

$x \circ_{\varepsilon} y = x \phi^{\varepsilon}(x^{-1} y)$

This a $\mathbb{Z}$ idempotent quasigroup. The most interesting case (relevant also for Definition 3 below) is the one when $\phi$ is an uniformly contractive automorphism of the topological group $G$. 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 4.  Carnot groups: these are a particular example of a conical group. The most trivial noncommutative Carnot group is the Heisenberg group.

Example 5. A group with an invertible self-mapping $\phi: G \rightarrow G$  such that $\phi(e) =e$, where $e$ is the identity of the group $G$. In this case the construction from Example 3 works here as well because there is no need for $\phi$ 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 $\Gamma$  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 $G$, with an identity preserving, locally defined invertible self map (as in Example 5) gives a $\mathbb{Z}$ local idempotent quasigroup.

Example 7. A particular case of Example 6, is a Lie group $G$ with the operations  defined for any $\varepsilon \in (0,+\infty)$ by

$x \circ_{\varepsilon} y = x \exp ( \varepsilon \log (x^{-1} y) )$

Example 8. A less symmetric example is the one of $X$ being a riemannian manifold, with associated operations  defined for any $\varepsilon \in (0,+\infty)$ by

$x \circ_{\varepsilon}y = \exp_{x}( \varepsilon \log_{x}(y))$

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 $\varepsilon \in \Gamma$) 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!

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I shall define now the operations of approximate sum and approximate difference associated to a $\Gamma$  idempotent quasigroup.

For any $\varepsilon \in \Gamma$, let use define $x \bullet_{\varepsilon} y = x \circ_{\varepsilon^{-1}} y$.

Definition 2.  The approximate sum operation is (for any $\varepsilon \in \Gamma$)

$\Sigma_{\varepsilon}^{x}(y,z) = x \bullet_{\varepsilon} ( (x \circ_{\varepsilon} y) \circ_{\varepsilon} z)$

The approximate difference operation is (for any $\varepsilon \in \Gamma$)

$\Delta_{\varepsilon}^{x}(y,z) = (x \circ_{\varepsilon} y) \bullet_{\varepsilon} (x \circ_{\varepsilon} z)$

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Suppose now that $X$ 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 $0$. We write $\varepsilon \rightarrow 0$ for a net in $\latex \Gamma$ which converges to the filter $0$. The image to have in mind is $\Gamma = (0, + \infty)$ with multiplication of reals as operation and with the filter $0$ as the filter generated by sets $(0, a)$ with $a> 0$. This filter is the restriction to the set $(0,\infty) \subset \Gamma$ of the filter of  neighbourhoods of the number $0 \in \mathbb{R}$.  Another example is $\Gamma = \mathbb{Z}$ with addition of integers as operation, seen as a discrete topological group, with the absolute generated by sets $\left\{ n \in \mathbb{Z} \, \, : \, \, n \leq M \right\}$ for all $M \in \mathbb{Z}$. For this example the neutral element (denoted by $1$ in Definition 1) is the integer $0$, therefore in this case we can change notations from multiplication to addition, $1$ becomes $0$, the absolute $0$ becomes $- \infty$ , and so on.

Definition 3. An emergent algebra (or uniform idempotent quasigroup) is a $\Gamma$ idempotent quasigroup  $X$, as in Definition 1, which satisfies the following topological conditions:

1. The family of operations $\circ_{\varepsilon}$ is compactly contractive, i.e. for any compact set $K \subset X$, for any $x \in K$ and for any open neighbourhood $U$ of $x$, there is an open set $A(K,U) \subset \Gamma$ which belongs to the absolute $0$ such that for any $u \in K$ and $\varepsilon \in A(K,U)$ we have $x \circ_{\varepsilon} u \in U$.
2. As $\varepsilon \rightarrow 0$ there exist the limits

$\lim_{\varepsilon \rightarrow 0} \Sigma^{x}_{\varepsilon} (y,z) = \Sigma^{x} (y,z)$  and $\lim_{\varepsilon \rightarrow 0} \Delta^{x}_{\varepsilon} (y,z) = \Delta^{x} (y,z)$

and moreover these limits are uniform with respect to $x,y,z$ in compact sets.

The structure theorem of emergent algebras is the following:

Theorem 1.  Let $X$ be  a $\Gamma$ emergent algebra. Then for any $x \in X$ the pair $(X, \Sigma^{x}(\cdot, \cdot))$ 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.