Does "in-between" multiplication preserve equality?

Question

In a magma $(S;*)$, multiplication on the left and the right preserves equality. That is, if $a=b$, then $c*a=c*b$ and $a*c=b*c$. But what about "in-between" multiplication? That is, if $a*c=b*c$, does it follow that $(a*d)*c=(b*d)*c$? Or, as I think is more likely, is there a magma where that property does not hold?

Answer 1

Let $a, b, c: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ be linear maps. Suppose that $c(x, y) = (x, 0)$. Suppose that $a(x, y) = (0, y), b(x, y) = (0, -y)$. Then $a\circ c = b\circ c = 0$. Let $d(x, y) = (y, x)$. Then $(a\circ d \circ c)(x,y) = (0, x)$ and $(b \circ d \circ c)(x, y) = (0, -x)$.