#2443 ⟨a, b | aaa=1, abbbb=b⟩

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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
  8. Anti-isomorphic instances

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.

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. b10 ⇒ b [5]
2. ab ⇒ b7 [4]
3. a3 ⇒ 1 [1]
# ab:aaa=1,abbbb=b b/a
bbbbbbbbbb=b
ab=bbbbbbb
aaa=1

Same cardinality

8 unique, 104 total

Σ#PresentationDescriptionRelated
91470⟨a, b | aaa=bb, abab=1⟩Finite non-Abelian group with 30 elements73 iso
91907⟨a, b | aab=a, bbbbb=1⟩Finite non-commutative monoid with 30 elements11 iso, 4 anti-iso
106560⟨a, b | aaa=a, bbbb=ab⟩Finite non-commutative monoid with 30 elements
107013⟨a, b | ab=aa, bbbbb=b⟩Finite non-commutative monoid with 30 elements
1111058⟨a, b | abba=bbb, baba=1⟩Finite non-Abelian group with 30 elements8 iso
1113115⟨a, b | bbb=aaa, aaba=b⟩Finite non-commutative monoid with 30 elements
1115979⟨a, b | aaa=aa, bbbb=ab⟩Finite non-commutative monoid with 30 elements
1120847⟨a, b | ab=aa, bbbbb=bb⟩Finite non-commutative monoid with 30 elements

Isomorphic instances

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

14 total

Σ#PresentationMapping
92594⟨a, b | aaa=1, bbbb=ab⟩φ(a) = aa, φ(b) = b
107509⟨a, b | aaa=1, aabbbb=b⟩φ(a) = aa, φ(b) = b
107521⟨a, b | aaa=1, ababbb=b⟩φ(a) = aa, φ(b) = b
107525⟨a, b | aaa=1, abbabb=b⟩φ(a) = aa, φ(b) = b
107527⟨a, b | aaa=1, abbbab=b⟩φ(a) = aa, φ(b) = b
108091⟨a, b | aaa=1, bbbb=aab⟩φ(a) = a, φ(b) = b
1122247⟨a, b | aaa=1, aabbbb=ab⟩φ(a) = a, φ(b) = b
1122304⟨a, b | aaa=1, baabbb=ab⟩φ(a) = a, φ(b) = b
1122308⟨a, b | aaa=1, bababb=ab⟩φ(a) = a, φ(b) = b
1122312⟨a, b | aaa=1, babbab=ab⟩φ(a) = a, φ(b) = b
1122319⟨a, b | aaa=1, bbaabb=ab⟩φ(a) = a, φ(b) = b
1122817⟨a, b | aaa=1, abbbb=aab⟩φ(a) = aa, φ(b) = b
1122845⟨a, b | aaa=1, babbb=aab⟩φ(a) = aa, φ(b) = b
1122853⟨a, b | aaa=1, bbabb=aab⟩φ(a) = aa, φ(b) = b

Anti-isomorphic instances

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

5 total

Σ#PresentationMapping
1122286⟨a, b | aaa=1, abbbba=ab⟩φ(a) = a, φ(b) = b
1122305⟨a, b | aaa=1, baabbb=ba⟩φ(a) = a, φ(b) = b
1122309⟨a, b | aaa=1, bababb=ba⟩φ(a) = a, φ(b) = b
1122818⟨a, b | aaa=1, abbbb=aba⟩φ(a) = aa, φ(b) = b
1122848⟨a, b | aaa=1, babbb=baa⟩φ(a) = aa, φ(b) = b