Alphabeta Math
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 ideals, filters, prime ideals and ultrafilters

Definition

Let B be a Boolean algebra. An ideal I⊆B contains 0, is downward closed, and is closed under binary joins. Thus b∈I and a≤b imply a∈I. A filter F⊆B contains 1, is upward closed, and is closed under binary meets. These definitions include the improper ideal and filter B.

An ideal is proper when 1∉I; a filter is proper when 0∉F. A prime ideal is a proper ideal such that a∧b∈I implies a∈I or b∈I. A prime filter is a proper filter such that a∨b∈F implies a∈F or b∈F.

An ultrafilter is a maximal proper filter under inclusion. A maximal ideal is a maximal proper ideal. Maximality always refers to proper objects. Consequently the trivial algebra has no prime ideals, maximal ideals, proper filters or ultrafilters. These are algebraic filters; a forcing preorder has a different filter definition.

The ideal convention here is downward closure. The upward inequality printed in Tressl 2.2.6(iv)(b) is not adopted.

Depends on

Dependency tree · 1 level

Sources