#4749 ⟨a, b | abba=b, baab=a⟩

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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.

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. a7 ⇒ a
  2. ba6 ⇒ b
  3. a3b ⇒ ba3
  4. b2 ⇒ a2ba2
  5. bab ⇒ aba
  6. ba2b ⇒ a 
# ab:abba=b,baab=a a/b
aaaaaaa=a
baaaaaa=b
aaab=baaa
bb=aabaa
bab=aba
baab=a

Right Cayley graph

Left Cayley graph

Others with same cardinality

7 unique, 13 total

Σ#PresentationDescriptionRelated
106381⟨a, b | aab=a, bbbbb=b⟩Finite non-commutative monoid with 25 elements1 iso
106536⟨a, b | aaa=a, aaba=bb⟩Finite non-commutative monoid with 25 elements
1111133⟨a, b | aaaaa=b, bbbbb=1⟩Isomorphic to ℤ253 iso
1114342⟨a, b | aaaa=a, aabba=b⟩Finite non-commutative monoid with 25 elements
1114869⟨a, b | abba=b, bbabb=a⟩Finite non-commutative monoid with 25 elements2 iso
1119634⟨a, b | aba=a, aaaaa=bb⟩Finite non-commutative monoid with 25 elements
1119706⟨a, b | aba=b, aaaaa=bb⟩Finite non-commutative monoid with 25 elements