#3645 ⟨a, b | aa=1, abab=bbb⟩

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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. b7 ⇒ b4 [4]
2. bab ⇒ ab3 [3]
3. a2 ⇒ 1 [1]
# ab:aa=1,abab=bbb b/a
bbbbbbb=bbbb
bab=abbb
aa=1

Same cardinality

5 unique, 7 total

Σ#PresentationDescriptionRelated
105088⟨a, b | aaa=bb, baab=a⟩Finite non-commutative monoid with 26 elements1 iso
107050⟨a, b | bb=aa, ababa=a⟩Finite non-commutative monoid with 26 elements
1115941⟨a, b | aba=bb, bbabb=a⟩Finite non-commutative monoid with 26 elements
1119766⟨a, b | aba=b, babbb=aa⟩Finite non-commutative monoid with 26 elements
1126639⟨a, b | aa=1, abbbbbbb=b⟩Finite non-commutative monoid with 26 elements1 iso

Isomorphic instances

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

12 total

Σ#PresentationMapping
93657⟨a, b | aa=1, abbb=bab⟩φ(a) = a, φ(b) = b
1010000⟨a, b | aa=1, ababab=bb⟩φ(a) = a, φ(b) = ab
1010321⟨a, b | aa=1, babab=abb⟩φ(a) = a, φ(b) = ab
1127541⟨a, b | aa=1, aaabab=bbb⟩φ(a) = a, φ(b) = b
1127555⟨a, b | aa=1, aaabbb=bab⟩φ(a) = a, φ(b) = b
1127619⟨a, b | aa=1, abaaab=bbb⟩φ(a) = a, φ(b) = b
1127631⟨a, b | aa=1, abaabb=bab⟩φ(a) = a, φ(b) = b
1127663⟨a, b | aa=1, abbaab=bab⟩φ(a) = a, φ(b) = b
1128099⟨a, b | aa=1, aabab=abbb⟩φ(a) = a, φ(b) = b
1128129⟨a, b | aa=1, aabbb=abab⟩φ(a) = a, φ(b) = b
1128230⟨a, b | aa=1, baaab=abbb⟩φ(a) = a, φ(b) = b
1128239⟨a, b | aa=1, baabb=abab⟩φ(a) = a, φ(b) = b

Anti-isomorphic instances

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

7 total

Σ#PresentationMapping
1127573⟨a, b | aa=1, aababa=bbb⟩φ(a) = a, φ(b) = b
1127601⟨a, b | aa=1, aabbba=bab⟩φ(a) = a, φ(b) = b
1128106⟨a, b | aa=1, aabab=bbba⟩φ(a) = a, φ(b) = b
1128134⟨a, b | aa=1, aabbb=baba⟩φ(a) = a, φ(b) = b
1128162⟨a, b | aa=1, ababa=abbb⟩φ(a) = a, φ(b) = b
1128202⟨a, b | aa=1, abbba=abab⟩φ(a) = a, φ(b) = b
1128244⟨a, b | aa=1, baabb=baba⟩φ(a) = a, φ(b) = b