---
tag: 71346N77AX
kind: definition
name: Entropic algebra
msc: [08A62, 20N02]
defines: 9T2XVJNBBY
uses: [QJGPT2W0FT, KBXSZC6BV1, WJ73XYBNE4, VZHN6GA3TA]
sources:
  - url: https://en.wikipedia.org/wiki/Medial_magma
---

## Definition

Let $A$ be an [[algebra|QJGPT2W0FT]].
An $n$-ary operation $f$ and an $m$-ary operation $g$ of $A$ *commute* if
$$f\bigl(g(a_{11}, \dots, a_{1m}), \dots, g(a_{n1}, \dots, a_{nm})\bigr) = g\bigl(f(a_{11}, \dots, a_{n1}), \dots, f(a_{1m}, \dots, a_{nm})\bigr)$$
for all elements $a_{ij} \in A$, $1 \le i \le n$, $1 \le j \le m$.
The algebra $A$ is *entropic* if every two of its operations commute, including each operation with itself.

## Remarks

Equivalently, $A$ is entropic if and only if each $n$-ary operation $f \colon A^n \to A$ is a [[homomorphism|KBXSZC6BV1]] from the [[direct power|WJ73XYBNE4]] $A^n$ to $A$.
For an algebra with a single binary operation, the condition for $f = g$ is exactly the [[medial law|VZHN6GA3TA]] $(xy)(uv) = (xu)(yv)$.

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 $y, u \in G$.