Median algebra
In mathematics, a median algebra is a set with a ternary operation ⟨ x , y , z ⟩ {\displaystyle \langle x,y,z\rangle } satisfying a set of axioms which generalise the notions of medians of triples of real numbers and of the Boolean majority function. The axioms are ⟨ x , y , y ⟩ = y {\displaystyle \langle x,y,y\rangle =y} ⟨ x , y , z ⟩ = ⟨ z , x , y ⟩ {\displaystyle \langle x,y,z\rangle =\langle z,x,y\rangle } ⟨ x , y , z ⟩ = ⟨ x , z , y ⟩ {\displaystyle \langle x,y,z\rangle =\langle x,z,y\rangle } ⟨ ⟨ x , w , y ⟩ , w , z ⟩ = ⟨ x , w , ⟨ y , w , z ⟩ ⟩ {\displaystyle \langle \langle x,w,y\rangle ,w,z\rangle =\langle x,w,\langle y,w,z\rangle \rangle } The second and third axioms imply commutativity: it is possible (but not easy) to show that in the presence of the other three, axiom (3) is redundant.