This question is a follow-up and analogue of a previous very similar question. I have to seperate questions since it is a bit of a rule ; 1 question at a time , hence this seperate one. Looking at the other analogue question is not neccessary and can be ignored but for those who care here it is :
About 2 operators and $A*(B+C) =(A+B)*(A+C)$
Anyway,
Consider $2$ distinct binary operators defined for a finite set with $n$ elements.
Operator $*$ behaves like a commutative latin quandle :
$$x*x = x$$
$$a*b=b*a$$
$$a*(b*c)=(a*b)*(a*c)$$
And forms a latin square.
Commutative operator $+$ behaves like
$$A*(B+C) =(A*B)+(A*C)$$
$$X+Y = Y+X$$
And forms a latin square.
Question :
Is $+$ neccessarily associative ?
What are examples of strict non-associative or strict associative ones if they exist ? ( strict mean always that property for every pair of elements )
The special cases where the quandle has no subquandle is perhaps interesting. ( this happens if $n$ is prime but that is not a neccessary condition , it is similar to subgroups basically )
ps: do not confuse with bi-quandles as most define them.
Remarks :
Also note the number of solutions $(*,+)$ for a set with $n$ elements is a bit of a mystery for me , Im not even sure with all these restrictions given that every $n$ has such a pair of operators. Not even for sufficiently large $n$.
But that is not the main question but just related. Unless there are no solutions $(*,+)$ for $n$ or only for finitely many $n$ then it becomes very important.