property of quandle

Question

Let $ Q $ be any quandle. Let $ y,k, w$ be elements of $ Q$. Is it true that if $ y*k=y*w $ then $ k=w $? I don't think so since the second axiom of the quandles states that for any two distinct elements $ x, y $ of a quandle, there is a unique element $ z $ such that $ x=z*y$. Suppose $ y*k = y*w=n $ this means $y$ is the unique element satisfies the condition above between $ n, k$ and also between $ n, w $. So there is nothing contradicts the axiom and we can not conclude that $ k=w $. Am I right.

Answer 1

Correct. Take the $3$-element quandle with multiplication table (faked with a matrix) $$\left[\begin{matrix}a&a&a\\c&b&b\\b&c&c \end{matrix}\right].$$

Then $a\ast b = a\ast a = a$, but $a\neq b$.

A quandle in which your property does happen to hold is said to be a latin quandle.