#26599 ⟨a, b | aa=1, ababbbba=b⟩

Up:Monoids with two generators and two relations
Prev:#26183 ⟨a, b | aa=1, bbbbbbbbb=1⟩
Next:#26601 ⟨a, b | aa=1, ababbbbb=b⟩

Properties

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The order of x is the least n (if it exists) such that xn = 1.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Complete rewriting system

Format:
Word to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 1 
1
  1. b9 ⇒ b
  2. b8ab ⇒ ab
  3. ab8 ⇒ b8a
  4. a2 ⇒ 1
  5. aba ⇒ bab4
  6. ab4a ⇒ b7ab
  7. ab7a ⇒ b4ab7
  8. ab3ab ⇒ b7ab5a
  9. ab6ab ⇒ b(b3a)2
  10. (ab2)2 ⇒ (b6a)2
  11. ab5ab4 ⇒ b7ab2a
  12. ab5ab2a ⇒ (b5a)2b
  13. ab5ab3a ⇒ (bab)2 
# ab:aa=1,ababbbba=b reversed:b/a
bbbbbbbbb=b
bbbbbbbbab=ab
abbbbbbbb=bbbbbbbba
aa=1
aba=babbbb
abbbba=bbbbbbbab
abbbbbbba=bbbbabbbbbbb
abbbab=bbbbbbbabbbbba
abbbbbbab=bbbbabbba
abbabb=bbbbbbabbbbbba
abbbbbabbbb=bbbbbbbabba
abbbbbabba=bbbbbabbbbbab
abbbbbabbba=babbab

Right Cayley graph

Left Cayley graph

Other isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

2 total

Σ#PresentationMapping
1127128⟨a, b | aa=1, ababbbb=ba⟩φ(a) = a, φ(b) = b
1127742⟨a, b | aa=1, babbbb=aba⟩φ(a) = a, φ(b) = b

Other anti-isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

1 total

Σ#PresentationMapping
1127159⟨a, b | aa=1, abbbbab=ba⟩φ(a) = a, φ(b) = b