Does there exist a magma where every element has a left cube root but not every element has a right cube root?

Question

Let $(M,*)$ be a magma. $x$ is said to be a left cube root of $y$ if $(x*x)*x=y$. $x$ is said to be a right cube root of $y$ if $x*(x*x)=y$. Does there exist a magma where every element has a left cube root, but not every element has a right cube root?

Answer 1

Consider the law (written without any sign): $$ \begin{array}{r|cc} & 0 & 1\\\hline 0 & 0 & 1\\ 1 & 0 & 0 \end{array} $$ Then: $$ \begin{aligned} (00)0 &= 00 = 0\ ,\\ (11)1 &= 01 =1\ ,\\[2mm] 0(00) &= 00 = 0\ ,\\ 1(11) &= 10 = 0\ . \end{aligned} $$