#2300 ⟨a, b | aaa=1, ababbb=1⟩

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Quick links

  1. Properties
  2. Elements
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. Same cardinality
  7. Isomorphic instances

Properties

Elements

Elements in the center commute with all other elements.
The order of x is the least n such that xn = 1.

Right Cayley graph

Left Cayley graph

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
#RuleProof
1. b12 ⇒ 1 [10]
2. b2a ⇒ ab2 [7]
3. a2 ⇒ bab3 [3]
4. aba ⇒ b9 [11]
# ab:aaa=1,ababbb=1 b/a
bbbbbbbbbbbb=1
bba=abb
aa=babbb
aba=bbbbbbbbb

Same cardinality

4 unique, 18 total

Σ#PresentationDescriptionRelated
91345⟨a, b | aaab=a, bbbb=1⟩Finite non-commutative monoid with 36 elements5 iso, 8 anti-iso
105073⟨a, b | aaa=ab, bbbb=b⟩Finite non-commutative monoid with 36 elements1 iso
1116044⟨a, b | aaa=ab, bbbb=bb⟩Finite non-commutative monoid with 36 elements
1118959⟨a, b | aab=a, bbbbbb=b⟩Finite non-commutative monoid with 36 elements

Isomorphic instances

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

23 total

Σ#PresentationMapping
92303⟨a, b | aaa=1, abbbab=1⟩φ(a) = a, φ(b) = b
92310⟨a, b | aaa=1, bababb=1⟩φ(a) = a, φ(b) = b
106946⟨a, b | bb=aa, ababab=1⟩φ(a) = abbb, φ(b) = b
107538⟨a, b | aaa=1, baabbb=a⟩φ(a) = abbbb, φ(b) = b
107814⟨a, b | aaa=1, babbb=aa⟩φ(a) = a, φ(b) = b
1113821⟨a, b | aaaa=a, ababbb=1⟩φ(a) = a, φ(b) = b
1113824⟨a, b | aaaa=a, abbbab=1⟩φ(a) = a, φ(b) = b
1113831⟨a, b | aaaa=a, bababb=1⟩φ(a) = a, φ(b) = b
1113960⟨a, b | aaab=b, ababbb=1⟩φ(a) = a, φ(b) = b
1113966⟨a, b | aaab=b, abbbab=1⟩φ(a) = a, φ(b) = b
1113980⟨a, b | aaab=b, bababb=1⟩φ(a) = a, φ(b) = b
1113983⟨a, b | aaab=b, babbba=1⟩φ(a) = a, φ(b) = b
1113990⟨a, b | aaab=b, bbabab=1⟩φ(a) = a, φ(b) = b
1113995⟨a, b | aaab=b, bbbaba=1⟩φ(a) = a, φ(b) = b
1121157⟨a, b | aaa=1, aabaabbb=1⟩φ(a) = abbbb, φ(b) = b
1121171⟨a, b | aaa=1, aabbbaab=1⟩φ(a) = abbbb, φ(b) = b
1121188⟨a, b | aaa=1, abaabbba=1⟩φ(a) = abbbb, φ(b) = b
1121228⟨a, b | aaa=1, baabaabb=1⟩φ(a) = abbbb, φ(b) = b
1121692⟨a, b | aaa=1, aababbb=a⟩φ(a) = a, φ(b) = b
1121702⟨a, b | aaa=1, aabbbab=a⟩φ(a) = a, φ(b) = b
1121724⟨a, b | aaa=1, abababb=a⟩φ(a) = a, φ(b) = b
1121728⟨a, b | aaa=1, ababbba=a⟩φ(a) = a, φ(b) = b
1121736⟨a, b | aaa=1, abbabab=a⟩φ(a) = a, φ(b) = b