Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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.

The injection ∅→{∗}\varnothing\to\{*\} is monic in Set\mathbf{Set} but is not split

Statement refuted

The assertion that every monomorphism is a split monomorphism is false.

Facts & Assumptions

Given: The unique function i:∅→{∗}i:\varnothing\to\{*\}.

[L1]

Monomorphisms in Set\mathbf{Set} are exactly injections (In Set\mathbf{Set}, monomorphisms are exactly injections and epimorphisms are exactly surjections).

[L2]

A split monomorphism i:A→Bi:A\to B requires a retraction r:B→Ar:B\to A with ri=1Ari=1_A (Split monomorphism, split epimorphism, retraction, and section).

Counterexample

technique · direct
1.1

The map ii is injective because its domain has no elements, so [L1] makes it a monomorphism.

L1
1.2

There is no function r:{∗}→∅r:\{*\}\to\varnothing, since the value r(∗)r(*) would have to be an element of the empty set.

given
2.1

Therefore ii admits no retraction and is not split by [L2], although it is monic by step 1.1.

step 1.1step 1.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 12 results over 5 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.