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.
Words in an alphabet with formal inverses, elementary cancellation, and reduced words
Definition
For a set , form a disjoint copy of formal inverses. A word on is a finite string of letters from ; the string of length zero is the empty word.
An elementary cancellation deletes two adjacent letters or . A word is reduced if no elementary cancellation applies. Words are freely equivalent if one can be transformed into the other by finitely many elementary cancellations and their reverse insertions. The reduction and uniqueness facts needed for the free-group construction are proved in Reduced words form the free group on an alphabet ↗.
Depends on
Used by
- With respect to a free basis, the word length of an element is the length of its reduced word Corollary
- Cyclically reduced words Definition
- Free group on a set of generators Definition
- Sc toolkit labelled planar disc diagram Definition
- Schreier transversals and Schreier systems Definition
- Singular planar labelled relator diagrams and their outer walks Definition
- The Schreier rewriting map Definition
- The word-quotient model F_word(X):=W(X)/∼ with multiplication induced by concatenation Definition
- Van Kampen diagrams, boundary labels, and diagram area for a presentation Definition
- Word length of a group element with respect to a generating set Definition
- Reducing the word xx⁻¹yy⁻¹x leaves the reduced word x Example
- A Cayley graph is connected if and only if the subset generates the group Lemma
- Every nonempty reduced word has the form tct⁻¹ with c nonempty and cyclically reduced Lemma
- Formal letters act by mutually inverse permutations on the set of reduced words Lemma
- Sc toolkit commuting positive words have a common root Lemma
- Schreier rewriting is invariant under free reduction Lemma
- The Schreier coset graph is connected and deterministic Lemma
- Free equivalence is an equivalence relation and concatenation respects it Proposition
- Every class in W(X)/∼ contains exactly one reduced word Theorem
- If no product of two members of a generating set is the identity and the Cayley graph is a tree, the set is a free basis Theorem
- Reduced words form the free group on an alphabet Theorem
- The Cayley graph of a free group with respect to a free basis is a tree Theorem
- The word problem for a finitely generated free group is solvable by free reduction Theorem
Dependency tree · two levels
6 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
- McKernan, Presentations and Groups of Small Order, Lecture 12 (standard reference, not scraped)