CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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A subobject lattice of an abelian category need not be distributive

Statement refuted

Every subobject lattice of an object in an abelian category is distributive.

Facts & Assumptions

Given: The abelian group $A = (Z / 2) \oplus (Z / 2)$ .

[L1]

Subobject lattices in an abelian category are modular (The subobject lattice of an abelian category is modular).

[L2]

A lattice is distributive when meet distributes over join (Lattices, distributive lattices, and order ideals).

Counterexample

1.1givenalgebra

The nonzero proper subgroups of $A$ are exactly the three one-dimensional subspaces $L_{1} = ⟨ (1, 0) ⟩, L_{2} = ⟨ (0, 1) ⟩, L_{3} = ⟨ (1, 1) ⟩ .$ For $i \neq j$ , the intersection $L_{i} \cap L_{j}$ is $0$ , and $L_{i} + L_{j} = A$ . So the subobject lattice of $A$ is the diamond $M_{3}$ .

2.1L1L2step 1.1∎

Now $L_{1} \land (L_{2} \lor L_{3}) = L_{1} \land A = L_{1},$ while $(L_{1} \land L_{2}) \lor (L_{1} \land L_{3}) = 0 \lor 0 = 0.$ Hence the distributive law of [L2] fails in this subobject lattice. By [L1], the example is modular but not distributive.

Depends on

Used by

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Sources

Daniel Murfet, Abelian Categories, Section 4.2 (standard reference, not scraped)