#7519 ⟨a, b | aaa=1, ababba=b⟩

Up:Monoids with two generators and two relations
Prev:#7279 ⟨a, b | aaa=1, bbbbbbb=1⟩
Next:#7529 ⟨a, b | aaa=1, abbbba=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. b15 ⇒ b
  2. bab14 ⇒ ba
  3. b7a ⇒ ab7
  4. ba2 ⇒ abab2
  5. (ba)2 ⇒ ab5ab3
  6. bab2a ⇒ a2b
  7. bab3a ⇒ ab3ab11
  8. bab5a ⇒ ab4ab6
  9. bab6a ⇒ ab2ab5
  10. b2ab4a ⇒ ab6ab4
  11. a3 ⇒ 1 
# ab:aaa=1,ababba=b b/a
bbbbbbbbbbbbbbb=b
babbbbbbbbbbbbbb=ba
bbbbbbba=abbbbbbb
baa=ababb
baba=abbbbbabbb
babba=aab
babbba=abbbabbbbbbbbbbb
babbbbba=abbbbabbbbbb
babbbbbba=abbabbbbb
bbabbbba=abbbbbbabbbb
aaa=1

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.

6 total

Σ#PresentationMapping
108084⟨a, b | aaa=1, babb=aba⟩φ(a) = a, φ(b) = abbab
1121717⟨a, b | aaa=1, abaabba=b⟩φ(a) = aa, φ(b) = abbab
1122233⟨a, b | aaa=1, aababb=ba⟩φ(a) = a, φ(b) = abbab
1122798⟨a, b | aaa=1, ababb=baa⟩φ(a) = a, φ(b) = b
1122806⟨a, b | aaa=1, abbab=baa⟩φ(a) = aa, φ(b) = aabbab
1122832⟨a, b | aaa=1, baabb=aba⟩φ(a) = aa, 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
1122240⟨a, b | aaa=1, aabbab=ba⟩φ(a) = a, φ(b) = abbab