Etude IX

Lattice Laws

Join, meet, monotonicity, and convergence guarantees.

Core Notes

Join-semilattice laws (commutative, associative, idempotent) guarantee convergence under reordering.

x ⊔ y = y ⊔ x, (x ⊔ y) ⊔ z = x ⊔ (y ⊔ z), x ⊔ x = x
Use set union as the canonical example; every update is information growth.

Applied Thread

Applied: verify merge laws per CRDT, then encode them as tests and runtime guards at the API boundary.

trace: Def → Check(laws) → Counterexample(non-idempotent) → Repair(redesign)