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.
I will also define a "linear kei" as structure generated by a set of 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.
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 answer
(This is about $\frac23$ of an answer.)
Firstly, to answer this question it may be helpful to make a little diagram of the different classes of quandles we're dealing with here:
And the equations satisfied by each of these classes of quandles/kei have inclusion relations going in the opposite way:
Then our question is to find whether any of the three inclusion relations on the left (among the different kinds of kei) is strict.
Joyce 1982 has shown that the relation on the right is non-strict: all equational laws satisfied by conjugation quandles are satisfied by quandles in general (pp. 40–41). So $eq_{\text{subquandles of conjugation quandles}} = eq_{\text{conjugation quandles}} = eq_{\text{quandles}}$.
Joyce's proof, which proceeds by showing that equalities in a free group correspond to equalities in a free quandle, appears straightforwardly generalizable to the case of kei, but I have not thought it out thoroughly.
Also, any subkei of a conjugation kei may be modeled as a hyperplanar kei, as any involution is a reflection around some hyperplane—so it's obvious $eq_{\text{subkei of conjugation kei}}$ equals $eq_{\text{hyperplanar kei}}$.
There remains the question whether linear, planar, etc. kei satisfy equational laws that hyperplanar kei don't. We can split this problem into two cases:
- Do planar, 3-planar, 4-planar, etc. kei satisfy the same equational laws as linear kei?
- Do hyperplanar kei in infinite-dimensional spaces satisfy the same equational laws as planar, 3-planar, 4-planar, etc. kei?
I suspect that both questions have a positive answer (the first question because planar, 3-planar, etc. reflections can be generated by linear reflections, and the second question because one should be able to construct the required infinite-dimensional kei by some kind of subdirect product construction), but I am not very sure.
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