#16044 ⟨a, b | aaa=ab, bbbb=bb⟩

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Quick links

  1. Properties
  2. Elements
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. Same cardinality

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. a9 ⇒ a5 [3]
2. ab ⇒ a3 [1]
3. b4 ⇒ b2 [2]
# ab:aaa=ab,bbbb=bb a/b
aaaaaaaaa=aaaaa
ab=aaa
bbbb=bb

Same cardinality

4 unique, 41 total

Σ#PresentationDescriptionRelated
91345⟨a, b | aaab=a, bbbb=1⟩Finite non-commutative monoid with 36 elements5 iso, 8 anti-iso
92300⟨a, b | aaa=1, ababbb=1⟩Finite non-Abelian group with 36 elements23 iso
105073⟨a, b | aaa=ab, bbbb=b⟩Finite non-commutative monoid with 36 elements1 iso
1118959⟨a, b | aab=a, bbbbbb=b⟩Finite non-commutative monoid with 36 elements