Median algebra

In mathematics, a median algebra is a set with a ternary operation ${\displaystyle \langle x,y,z\rangle }$ satisfying a set of axioms which generalise the notion of median or majority function, as a Boolean function.

The axioms are

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. The fourth axiom implies associativity. There are other possible axiom systems: for example the two

also suffice.

In a Boolean algebra, or more generally a distributive lattice, the median function ${\displaystyle \langle x,y,z\rangle =(x\vee y)\wedge (y\vee z)\wedge (z\vee x)}$ satisfies these axioms, so that every Boolean algebra and every distributive lattice forms a median algebra.

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