Post History
#4: Post edited
by
clemens
·
2026-05-21T18:51:52Z (4 months ago)
clarified and expanded last question
- (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](https://www.sciencedirect.com/science/article/pii/0022404982900779?via%3Dihub) 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:
- 1. Do planar, 3-planar, 4-planar, etc. kei satisfy the same equational
laws as hyperplanar kei?- 2. 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.
- (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](https://www.sciencedirect.com/science/article/pii/0022404982900779?via%3Dihub) 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:
- 1. Do planar, 3-planar, 4-planar, etc. kei satisfy the same equational
- laws as linear kei?
- 2. 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.
#3: Post edited
by
clemens
·
2026-05-21T18:51:38Z (4 months ago)
clarified and expanded last question
Not quite an answer, merely some (useful, I hope) pointers:- 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.
It is easy to show that *one* of the relations is not strict: any subkei of a conjugation kei may be modeled as a hyperplanar kei, as any involution is a reflection around some hyperplane—so $eq_{\text{subkei of conjugation kei}}$ clearly equals $eq_{\text{hyperplanar kei}}$.Less trivially, [Joyce 1982](https://www.sciencedirect.com/science/article/pii/0022404982900779?via%3Dihub) 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.
There remains only the question whether linear, planar, etc. kei satisfy equational laws that hyperplanar kei don't. I expect that it can be proven that they don't, because hyperplanar reflections can be generated by linear reflections. But I am not fully sure of this either.
- (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](https://www.sciencedirect.com/science/article/pii/0022404982900779?via%3Dihub) 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:
- 1. Do planar, 3-planar, 4-planar, etc. kei satisfy the same equational
- laws as hyperplanar kei?
- 2. 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.
#2: Post edited
by
clemens
·
2026-05-20T03:01:37Z (4 months ago)
- Not quite an answer, merely some (useful, I hope) pointers:
- 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.
- It is easy to show that *one* of the relations is not strict: any subkei of a conjugation kei may be modeled as a hyperplanar kei, as any involution is a reflection around some hyperplane—so $eq_{\text{subkei of conjugation kei}}$ clearly equals $eq_{\text{hyperplanar kei}}$.
- Less trivially, [Joyce 1982](https://www.sciencedirect.com/science/article/pii/0022404982900779?via%3Dihub) 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.
There remains the question whether linear, planar, etc. kei satisfy equational laws that hyperplanar kei don't. I expect that it can be proven that they don't, because hyperplanar reflections can be generated by linear reflections. But I am not fully sure of this either.
- Not quite an answer, merely some (useful, I hope) pointers:
- 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.
- It is easy to show that *one* of the relations is not strict: any subkei of a conjugation kei may be modeled as a hyperplanar kei, as any involution is a reflection around some hyperplane—so $eq_{\text{subkei of conjugation kei}}$ clearly equals $eq_{\text{hyperplanar kei}}$.
- Less trivially, [Joyce 1982](https://www.sciencedirect.com/science/article/pii/0022404982900779?via%3Dihub) 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.
- There remains only the question whether linear, planar, etc. kei satisfy equational laws that hyperplanar kei don't. I expect that it can be proven that they don't, because hyperplanar reflections can be generated by linear reflections. But I am not fully sure of this either.
#1: Initial revision
by
clemens
·
2026-05-20T03:01:25Z (4 months ago)
Not quite an answer, merely some (useful, I hope) pointers:
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.
It is easy to show that *one* of the relations is not strict: any subkei of a conjugation kei may be modeled as a hyperplanar kei, as any involution is a reflection around some hyperplane—so $eq_{\text{subkei of conjugation kei}}$ clearly equals $eq_{\text{hyperplanar kei}}$.
Less trivially, [Joyce 1982](https://www.sciencedirect.com/science/article/pii/0022404982900779?via%3Dihub) 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.
There remains the question whether linear, planar, etc. kei satisfy equational laws that hyperplanar kei don't. I expect that it can be proven that they don't, because hyperplanar reflections can be generated by linear reflections. But I am not fully sure of this either.