CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

How 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.

Algebras for a preorder monad are exactly its fixed objects up to preorder equivalence; on a poset they are its fixed points

Statement

For a monad $T$ on a preorder $P$ , a $T$ -algebra structure exists on $p$ exactly when $p$ and $T p$ are mutually comparable. On a poset this says exactly that $T p = p$ .

Facts & Assumptions

Given: A monad $T$ on a preorder $P$ .

[L1]

A $T$ -algebra on $p$ includes an arrow $T p \to p$ (Algebra and algebra homomorphism for a monad).

[L2]

The monad unit supplies $p \leq T p$ , and every required diagram between fixed objects of a preorder commutes automatically (On a preorder the monads are exactly the monotone extensive maps with T(Tp) below Tp; on a poset they are exactly the closure operators).

[L3]

Antisymmetry turns mutual comparability into equality (Partial order and partially ordered set).

Proof

technique · direct

1.1L1L2

If $p$ carries a $T$ -algebra, [L1] gives $T p \leq p$ , while [L2] gives $p \leq T p$ ; hence the two objects are mutually comparable.

2.1L1L2step 1.1

Conversely, if $T p \leq p$ , the corresponding unique arrow is an algebra structure: its unit and associativity diagrams commute because parallel arrows in a preorder are equal.

3.1L3step 2.1∎

When $P$ is a poset, [L3] changes mutual comparability into $T p = p$ , and equality plainly gives the comparison required in step 2.1.

Depends on

Used by

Topological closure is a monad on the preorder of subsets Example

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, 2nd ed., Examples 5.1.7 and 5.2.6(iv) (standard reference, not scraped)