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Q&A Is there an equational law satisfied by "linear kei" but not free kei?

0 answers · posted 3mo ago by WheatWizard‭ · last activity 2mo ago by celtschk‭

Question geometry abstract-algebra

#2: Post edited by $user avatar$ WheatWizard‭ · 2025-01-06T15:52:55Z (2 months ago)
Clarified two things. First that a linear kei need not contain all the lines in n-space, next that I am looking for an equational law.

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~~Is there a law satisfied by "linear kei" but not free kei?~~

Is there an equational law satisfied by "linear kei" but not free kei?

A kei is an involutionary quandle, alternatively a magma satisfying three equations:
* $a ⊳ a = a$
* $(a ⊳ b) ⊳ b = a$
* $(a ⊳ b) ⊳ c = (a ⊳ c) ⊳ (b ⊳ c)$
This definition is taken from [Kamada, S., 2002](https://msp.org/gtm/2002/04/p008.xhtml).
I will also define a "linear kei" as structure generated lines in $R^{n}$ with the operator $a h d b$ defined as reflection of $a$ over $b$ . It is easy enough to verify the three laws above hold for this operation, and thus linear kei are kei.
**My question is whether there exists some equational law which is satisfied by all linear kei but does not hold for all kei.**
~~As motivation, kei generated by point reflections satisfy the law~~
\begin{equation}
a \rhd (b \rhd c) = (a \rhd b) \rhd (a \rhd c)
\end{equation}
But it does not hold for kei generated by linear reflections. As the following diagram demonstrates:
![Counter example of right distribution for linear kei](https://math.codidact.com/uploads/5mieazzdnukzezkglk09heteg65j)

A kei is an involutionary quandle, alternatively a magma satisfying three equations:
* $a ⊳ a = a$
* $(a ⊳ b) ⊳ b = a$
* $(a ⊳ b) ⊳ c = (a ⊳ c) ⊳ (b ⊳ c)$
This definition is taken from [Kamada, S., 2002](https://msp.org/gtm/2002/04/p008.xhtml).
I will also define a "linear kei" as structure generated by a set of lines in $R^{n}$ with the operator $a h d b$ defined as reflection of $a$ over $b$ . It is easy enough to verify the three laws above hold for this operation, and thus linear kei are kei.
**My question is whether there exists some equational law which is satisfied by all linear kei but does not hold for all kei.**
I mean equational law in the sense of universal algebra. It consists of some number of universally quantified variables, and two expressions in terms of the kei operation connected by equality.
As motivation, kei generated by point reflections satisfy the equational law
\begin{equation}
a \rhd (b \rhd c) = (a \rhd b) \rhd (a \rhd c)
\end{equation}
But it does not hold for kei generated by linear reflections. As the following diagram demonstrates:
![Counter example of right distribution for linear kei](https://math.codidact.com/uploads/5mieazzdnukzezkglk09heteg65j)

#1: Initial revision by $user avatar$ WheatWizard‭ · 2025-01-03T19:00:51Z (3 months ago)

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Is there a law satisfied by "linear kei" but not free kei?

A kei is an involutionary quandle, alternatively a magma satisfying three equations:

* $a \rhd a = a$
* $(a \rhd b) \rhd b = a$
* $(a \rhd b) \rhd c = (a \rhd c) \rhd (b \rhd c)$

This definition is taken from [Kamada, S., 2002](https://msp.org/gtm/2002/04/p008.xhtml).

I will also define a "linear kei" as structure generated lines in $\mathbb R^n$ with the operator $a \rhd b$ defined as reflection of $a$ over $b$. It is easy enough to verify the three laws above hold for this operation, and thus linear kei are kei.

**My question is whether there exists some equational law which is satisfied by all linear kei but does not hold for all kei.**

As motivation, kei generated by point reflections satisfy the law

\begin{equation}
a \rhd (b \rhd c) = (a \rhd b) \rhd (a \rhd c)
\end{equation}

But it does not hold for kei generated by linear reflections. As the following diagram demonstrates:

![Counter example of right distribution for linear kei](https://math.codidact.com/uploads/5mieazzdnukzezkglk09heteg65j)

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