Entropic algebra

Remarks

Equivalently, $A$ is entropic if and only if each $n$-ary operation $f \colon A^n \to A$ is a homomorphism from the direct power $A^n$ to $A$. For an algebra with a single binary operation, the condition for $f = g$ is exactly the medial law $(xy)(uv) = (xu)(yv)$.

Example: a module $M$ over a commutative ring $R$, with addition, the constant $0$, negation and one unary operation $x \mapsto rx$ for each $r \in R$. Addition commutes with itself because $(a + b) + (c + d) = (a + c) + (b + d)$. Scalar multiplication commutes with addition because $r(a + b) = ra + rb$, and two scalars commute because $r(sa) = (rs)a = (sr)a = s(ra)$. The checks for $0$ and negation, such as $r0 = 0$ and $-(a + b) = (-a) + (-b)$, are similar.

Non-example: a nonabelian group $G$ under multiplication. Multiplication commutes with itself only if $(xy)(uv) = (xu)(yv)$ for all $x, y, u, v$; with $x = v = 1$ this gives $yu = uy$, which fails for some $y, u \in G$.