#14869 ⟨a, b | abba=b, bbabb=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. b13 ⇒ b
  2. ab12 ⇒ a
  3. ba ⇒ ab5
  4. a2 ⇒ b3 
# ab:abba=b,bbabb=a b/a
bbbbbbbbbbbbb=b
abbbbbbbbbbbb=a
ba=abbbbb
aa=bbb

Right Cayley graph

Left Cayley graph

Others with same cardinality

7 unique, 11 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
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
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

Other isomorphic instances

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

2 total

Σ#PresentationMapping
1119218⟨a, b | aba=b, baabbb=a⟩φ(a) = bb, φ(b) = a
1119222⟨a, b | aba=b, babbab=a⟩φ(a) = bb, φ(b) = a