#5088 ⟨a, b | aaa=bb, baab=a⟩

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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.
An idempotent element x satisfies x2 = x.
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. a13 ⇒ a [4]
2. ab ⇒ ba5 [3]
3. b2 ⇒ a3 [1]
# ab:aaa=bb,baab=a a/b
aaaaaaaaaaaaa=a
ab=baaaaa
bb=aaa

Same cardinality

5 unique, 25 total

Σ#PresentationDescriptionRelated
93645⟨a, b | aa=1, abab=bbb⟩Finite non-commutative monoid with 26 elements12 iso, 7 anti-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.

1 total

Σ#PresentationMapping
107051⟨a, b | bb=aa, ababa=b⟩φ(a) = b, φ(b) = baa