Post History
#2: Post edited
by
WheatWizard
·
2025-01-06T15:52:55Z (1 day 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.
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:
- * $a \rhd a = a$
- * $(a \rhd b) \rhd b = a$
- * $(a \rhd b) \rhd c = (a \rhd c) \rhd (b \rhd c)$
- 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 $a hd b$ 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:
- 
- 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 by a set of lines in $\mathbb R^n$ with the operator $a hd b$ 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:
- 
#1: Initial revision
by
WheatWizard
·
2025-01-03T19:00:51Z (4 days ago)
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:
