Median algebra

Remarks

Clauses 2 and 3 together say that $\langle x, y, z \rangle$ does not change under any permutation of $x, y, z$. Clause 4 says that for fixed $w$, the binary operation $(x, y) \mapsto \langle x, w, y \rangle$ is associative.

Example: $X = \R$ with $\langle x, y, z \rangle$ the median of $x, y, z$, which equals $\max(\min(x, y), \min(y, z), \min(z, x))$. More generally, every distributive lattice is a median algebra with $\langle x, y, z \rangle = (x \wedge y) \vee (y \wedge z) \vee (z \wedge x)$. For clause 1, $(x \wedge y) \vee y \vee (y \wedge x) = y$.

Non-example: a set $X$ with at least two elements and $\langle x, y, z \rangle = x$. Clause 1 fails: for $x \ne y$, $\langle x, y, y \rangle = x \ne y$.