Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-29
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.

A general member of an abelian group need not come from an element

Statement refuted

Every member of an abelian group is equivalent to one arising from an ordinary element, that is, from a morphism Z→A.

Facts & Assumptions

Given: The identity member 1Z2:Z2→Z2.

[L1]

Member classes correspond to subobjects (Members modulo equivalence correspond to subobjects).

[L2]

The category Ab is abelian (Abelian groups form an abelian category).

Counterexample

technique · direct
1.1L1L2

The image of the member 1Z2 is all of Z2. By [L1], its equivalence class corresponds to the whole subgroup Z2≤Z2.

2.1L1step 1.1algebra

Any member coming from a map Z→Z2 has cyclic image, because the image of Z is generated by the image of 1. The subgroup Z2 is not cyclic. Therefore 1Z2 is not equivalent to any member Z→Z2.

3.1step 2.1∎

This refutes the statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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