ordered-locale

npx skills add majiayu000/claude-skill-registry --skill ordered-locale-proper --agent claude-code

Same command for any agent — swap --agent for codex, cursor, copilot.

# Ordered Locale Skill > *"We extend Stone duality between topological spaces and locales to include order."* > — Heunen & van der Schaaf, 2024 ## Overview An **ordered locale** is a locale (point-free topological space) equipped with a compatible preorder satisfying the **open cone condition**. ``` Ordered Locale = Frame + Preorder + Open Cones ``` ## Key Definitions ### Frame (Locale) A **frame** is a complete lattice where finite meets distribute over arbitrary joins: ``` a ∧ (⋁ᵢ bᵢ) = ⋁ᵢ (a ∧ bᵢ) ``` Equivalently: a complete Heyting algebra. ### Open Cone Condition For a preorder ≤ on locale L, the **open cone condition** requires: ``` ↑x = {y ∈ L : x ≤ y} is an open in L (upper cone) ↓x = {y ∈ L : y ≤ x} is an open in L (lower cone) ``` This ensures the order is "visible" to the topology. ### Ordered Locale (Definition 2.1, Heunen-van der Schaaf) An **ordered locale** is a tuple (L, ≤) where: 1. L is a locale (frame of opens) 2. ≤ is a preorder on O(L) 3. The open cone condition holds ## Stone Duality Extended ``` ┌────────────────────────┐ adjunction ┌─────────────────────┐ │ Preordered Topological │ ←───────────────→ │ Ordered Locales │ │ Spaces (open cones) │ │ (spatial) │

Run ordered locale demo
just ordered-locale-demo
Verify open cone condition
just ordered-locale-verify
Scale test with MLX
just ordered-locale-mlx