Medial magma

Remarks

Example: $\Q$ with $x \circ y = (x + y)/2$. Both $(x \circ y) \circ (u \circ v)$ and $(x \circ u) \circ (y \circ v)$ equal $(x + y + u + v)/4$. This magma is medial but not associative.

Example: every commutative semigroup is medial, since $(xy)(uv) = x(yu)v = x(uy)v = (xu)(yv)$.

Non-example: $\Z$ with $x \cdot y = xy + 1$. Take $x = y = 1$ and $u = v = 0$. Then $1 \cdot 1 = 2$ and $0 \cdot 0 = 1$, so the left side is $2 \cdot 1 = 3$. Also $1 \cdot 0 = 1$, so the right side is $1 \cdot 1 = 2 \ne 3$.

A magma is medial if and only if the map $M \times M \to M$, $(x, y) \mapsto x \cdot y$, is a homomorphism from the product magma $M \times M$ to $M$.