Median Algebra Explorer: Ternary Compromises & Graph Geodesics

Structure Presets:

Selected Triple: x: A y: B z: C → Median m(x,y,z): D Click nodes to set x, y, or z

Node x

Node y

Node z

Unique Median m

Computed Median

Pairwise Distance Sum

d(x,m) + d(y,m) + d(z,m)

Ternary Status

Unique Compromise

Axiomatic Foundations

Ternary Algebra (Birkhoff-Kiss)

1. Majority / Idempotence

m(x, x, y) = x & m(x, y, x) = x

✓ Verified

2. Full Permutation Symmetry

m(x,y,z) = m(y,x,z) = m(x,z,y)

✓ Invariant

3. Distributive Property

m(m(x,w,y), u, z) = m(m(x,u,z), w, m(y,u,z))

✓ Satisfied

4. Geodesic Intersection

I(x,y) ∩ I(y,z) ∩ I(x,z) = {m}

✓ Unique {D}

Shortest-Path Distance Matrix d(u,v)

Ternary Computation Log

Theoretical Context: A median algebra is an algebraic system satisfying ternary consensus rules. In every median graph (such as trees and hypercubes), the ternary operation $m(x,y,z)$ corresponds exactly to the unique node lying simultaneously on geodesics between all pairs.