Median algebra as a ternary algebra

Example

A median algebra is a set $A$ with one ternary operation $m$, written $\langle x, y, z \rangle$, such that for all $x, y, z, w \in A$:

1. $\langle x, y, y \rangle = y$;
2. $\langle x, y, z \rangle = \langle z, x, y \rangle$;
3. $\langle x, y, z \rangle = \langle x, z, y \rangle$;
4. $\langle \langle x, w, y \rangle, w, z \rangle = \langle x, w, \langle y, w, z \rangle \rangle$.

So it is an algebra with a single operation of arity $3$, defined by identities.

The basic instance is a distributive lattice $(L, \wedge, \vee)$ with $$\langle x, y, z \rangle = (x \wedge y) \vee (y \wedge z) \vee (z \wedge x).$$ On a total order this is the middle one of the three elements. For instance on $\Z$ with the usual order, $\langle 1, 5, 3 \rangle = 3$ and $\langle 4, 4, 9 \rangle = 4$. On $\set{0, 1}$ it is the majority function, and on $\set{0, 1}^n$ it is the majority taken in each coordinate. Laws 1 to 3 are clear for the majority function, and law 4 can be checked on $\set{0, 1}$ by going through the cases.