Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Galois connection between preorders
Definition
Let and be preorders (Preorder and monotone map). A Galois connection consists of monotone maps and such that, for every and ,
Under the identification of preorders with thin categories in A preorder is a category with at most one morphism between any two objects, and its functors are exactly monotone maps, this is exactly an adjunction. The corresponding unit and counit are the inequalities
Depends on
Used by
- A Galois connection between posets satisfies FGF=F and GFG=G Corollary
- A floor-division and multiplication adjunction between natural-number preorders Example
- Ceiling dashv inclusion dashv floor: an adjoint triple between (ℝ,≤) and (ℤ,≤) Example
- In a poset adjunction the triangle identities are automatic Proposition
- Direct image, preimage, and universal image form an adjoint triple on power sets Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 6 results over 3 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Emily Riehl, Category Theory in Context, 2nd ed., Section 4.2 (standard reference, not scraped)
- Tom Leinster, Basic Category Theory, Example 2.2.7 (standard reference, not scraped)