DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Boolean homomorphisms and quotient relation

Definition

A Boolean homomorphism $h : B \to C$ between Boolean algebras preserves $0, 1, \land, \lor, \neg$ . An isomorphism is a bijective homomorphism. Define Boolean symmetric difference by

$a △ b = (a \land \neg b) \lor (b \land \neg a) .$

For an ideal $I$ of $B$ , put $a \sim_{I} b$ when $a △ b \in I$ . The proposed quotient $B / I$ consists of the equivalence classes $[a]$ , with proposed operations

$[a] \land [b] = [a \land b], [a] \lor [b] = [a \lor b], \neg [a] = [\neg a], 0 = [0], 1 = [1] .$

The equivalence, representative independence and Boolean laws are justified by Quotient operations are well defined ↗. No representatives are selected as a family. The ideal $I = B$ is allowed; its quotient will be trivial. The kernel of a homomorphism means its zero fibre $h^{- 1} ({0})$ .

Depends on

Used by

Completeness, regular opens, and order continuity Definition
A finite quotient and its dual inclusion Example
Generated filters and the complementary-pair tests Lemma
Quotient operations are well defined Lemma
Stone duality and its natural maps Theorem

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

Tressl, Stone Duality for Boolean Algebras, Definition 2.3.1, pp. 8–9 (homomorphisms) (standard reference, not scraped)