ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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Adjoining one of two fixed points defines commuting closure-operator monads whose distributive law yields their composite

Example

On $P ({a, b, c})$ ordered by inclusion, define

$S (A) = A \cup {a}, T (A) = A \cup {b} .$

These closure-operator monads commute, and their equality $S T = T S$ gives a distributive law whose composite adjoins both $a$ and $b$ .

Facts & Assumptions

Given: The inclusion poset $P = P ({a, b, c})$ and the displayed maps $S, T : P \to P$ .

[L1]

A monotone, extensive, idempotent map on a poset defines a monad (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).

[L2]

A distributive law $S T \Rightarrow T S$ must satisfy the unit and multiplication compatibility diagrams (Distributive law between two monads).

[L3]

Such a distributive law gives a monad structure on $T S$ (A distributive law makes the composite endofunctor a monad).

Verification

technique · direct

1.1

Union with a fixed subset is monotone, extensive, and idempotent. Thus $S$ and $T$ are closure-operator monads by [L1].

2.1

For every $A$ , $S (T (A)) = A \cup {a, b} = T (S (A))$ . This equality gives $λ : S T \Rightarrow T S$ ; all diagrams in [L2] commute because $P$ is thin and the parallel arrows have the displayed common endpoints.

L2step 1.1

3.1

On $\emptyset, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}$ , both composites respectively give ${a, b}, {a, b}, {a, b}, {a, b, c}, {a, b}, {a, b, c}, {a, b, c}, {a, b, c}$ . This checks the formula at every object, including the empty set.

step 2.1

4.1

By [L3], the distributive law makes $T S$ a monad; its unit is the inclusion $A \subseteq A \cup {a, b}$ and its multiplication is the idempotence equality for the closure operator adjoining both fixed points.

L1L3step 2.1step 3.1∎

Depends on

Used by

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