Alphabeta MathHow 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.
Reduced syllable words in a family of groups
Definition
For groups as in Group and abelian group, a syllable is a tagged pair with and . A reduced syllable word is a finite list of syllables, indexed by a natural length as in The natural numbers (von Neumann), in which adjacent tags differ. The empty list is allowed. At a concatenation seam, adjacent syllables from the same factor are multiplied and an identity result is deleted; this elementary reduction is repeated until the seam is reduced.
Depends on
Used by
- Finite-order elements of a free product are conjugate into factors Corollary
- The center of a free product with at least two nontrivial factors is trivial Corollary
- Factor elements act by mutually inverse permutations on reduced syllable words Lemma
- Normal form theorem for free products Theorem
- Reduced syllable words form the free product of a family of groups Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 21 results over 11 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.
Sources
- George D. Torres, Combinatorial Group Theory, §2 (standard reference, not scraped)
- B. H. Neumann, Lectures on Topics in the Theory of Infinite Groups, Ch. 9 (standard reference, not scraped)