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median algebra

Median algebra is an algebra with a single ternary operation satisfying a set of three axioms. This is largely a generalized concept of the majority function.

Axioms

For majority function Equation f colon double-struck upper B cubed right-arrow double-struck upper B with Equation w comma x comma y comma z element-of double-struck upper B :

Name Axiom
Majority Equation Maj left-parenthesis x comma x comma y right-parenthesis equals x
Commutative Equation Maj left-parenthesis x comma y comma z right-parenthesis equals Maj left-parenthesis x comma z comma y right-parenthesis equals Maj left-parenthesis y comma x comma z right-parenthesis equals Maj left-parenthesis y comma z comma x right-parenthesis equals Maj left-parenthesis z comma x comma y right-parenthesis equals Maj left-parenthesis z comma y comma x right-parenthesis
Associative Equation Maj left-parenthesis x comma w comma Maj left-parenthesis y comma w comma z right-parenthesis right-parenthesis equals Maj left-parenthesis Maj left-parenthesis x comma w comma y right-parenthesis comma w comma z right-parenthesis


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