Closure Operators as API Contracts

Think of every endpoint as computing a closure: a generator produces candidates, a law L is applied until stable, and the fixed point is returned. Declaring “Endpoint : Generator → FixedPoint<S, L>” tells clients the input/output shape and guarantees:

• Soundness: output is closed under L (apply L again, no change).
• Minimality: nothing extra beyond what L forces from the seed.
• Monotonicity: larger seeds yield larger (never weaker) fixed points, enabling incremental recompute/cache.
• Idempotence: re-run returns the same fixed point.

Reusable template:

• Name: e.g., transitive-closure
• Law L: “extend by relation R until stable”
• Input (Generator): seed set S
• Output (Fixed Point): least X ⊇ S with “if a R b and a ∈ X then b ∈ X”
• Properties: idempotent, monotone, inflationary, least fixed point
• Observables: iterations to convergence, per-step delta (for SSE), stable hash for caching.

Examples:

• Graph reachability: seed nodes → follow edges to saturation.
• Document normalization: raw doc → apply rewrites until stable.
• Authorization derivation: asserted roles → expand via inheritance until stable.

Verification is light: check idempotence, monotonicity, minimality (remove any element and closure breaks). Composition becomes mechanical: closures over compatible laws compose; distributing laws can swap order safely, turning pipelines into algebra you can rearrange without changing meaning.