#6536 ⟨a, b | aaa=a, aaba=bb⟩

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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. a3 ⇒ a
  2. b2 ⇒ a2ba
  3. ba2ba ⇒ a(ab)2
  4. (ab)2a2 ⇒ (ab)2
  5. (ab)3 ⇒ (ab)2 
# ab:aaa=a,aaba=bb reversed:a/b
aaa=a
bb=aaba
baaba=aabab
ababaa=abab
ababab=abab

Right Cayley graph

Left Cayley graph

Others with same cardinality

7 unique, 13 total

Σ#PresentationDescriptionRelated
104749⟨a, b | abba=b, baab=a⟩Finite non-commutative monoid with 25 elements
106381⟨a, b | aab=a, bbbbb=b⟩Finite non-commutative monoid with 25 elements1 iso
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