home / skills / plurigrid / asi / join-semilattice

join-semilattice skill

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This skill helps you model join semilattices and preserve sup principles within data-driven updates across components and algorithms.

npx playbooks add skill plurigrid/asi --skill join-semilattice

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Files (2)

SKILL.md

812 B

---
name: join-semilattice
description: "Join semilattices and sup-preservation."
metadata:
  letter: J
  trit: -1
  seed: 1143
---
# J: Join Semilattice

> *Join semilattices and sup-preservation.*

## Directory Tree

```
join-semilattice/
├── SKILL.md
├── order/
├── lattice.idr
```

## GF(3) Assignment

Letter J → trit = -1

## Integration

```scheme
(define (join_semilattice x)
  ;; Join semilattices and sup-preservation.
  x)
```


---

## Autopoietic Marginalia

> **The interaction IS the skill improving itself.**

Every use of this skill is an opportunity for worlding:
- **MEMORY** (-1): Record what was learned
- **REMEMBERING** (0): Connect patterns to other skills  
- **WORLDING** (+1): Evolve the skill based on use



*Add Interaction Exemplars here as the skill is used.*

Overview

This skill implements and documents core concepts around join semilattices and sup-preservation. It provides a compact reference and small runtime integration point for working with joins, directed sups, and related order-theoretic operations. It is intended for engineers and researchers who need a lightweight, composable building block for algebraic ordering in proofs or systems.

How this skill works

The skill inspects algebraic structures that support a binary join operation and checks behavior relevant to sup-preservation: whether joins preserve existing suprema for relevant subsets. It exposes a minimal integration function that returns the structure unchanged while serving as a clear hook for embedding checks or transformation logic. The implementation can be extended to validate identities (associativity, commutativity, idempotence) and test sup-preservation properties against finite samples.

When to use it

Modeling or reasoning about partial orders where binary join is the primary operation.
Verifying that a transformation or mapping preserves suprema (sup-preserving homomorphisms).
Embedding order-theoretic constraints into proof assistants, DSLs, or type-level code.
Quick validation of algebraic properties during unit tests or property-based testing.
Teaching or demonstrating join semilattice behavior with minimal code.

Best practices

Always verify the three join identities: associativity, commutativity, and idempotence before testing sup-preservation.
Test sup-preservation on representative finite subsets and on edge cases like empty sets and singletons.
Use the provided integration hook to insert logging or property checks rather than mutating structures directly.
Keep the join operation total on the domain you model; explicitly document any partiality.
Compose small verified components: prefer simple semilattice modules over monolithic order code.

Example use cases

Asserting that a map between two ordered types is sup-preserving during a refactor of an abstract interpreter.
Unit tests that confirm a constructed join operation on a custom datatype satisfies lattice identities.
Embedding a join semilattice module in a DSL that requires deterministic merge semantics for concurrent updates.
Teaching exercises: students implement join and then validate sup-preservation with supplied test cases.
Rapid prototyping of algebraic kernels used in synthesis tools or symbolic computation pipelines.

FAQ

What exactly does sup-preserving mean here?

A function is sup-preserving if it maps the supremum (least upper bound) of a set to the supremum of the image of that set. Practically, check that f(sup S) = sup f(S) for representative S.

Can this handle infinite sups?

The core is designed for finite, concrete checks. Reasoning about infinite sups requires domain-specific completeness proofs or theorem-prover support.