#15979 ⟨a, b | aaa=aa, bbbb=ab⟩

Up:Monoids with two generators and two relations
Prev:#15978 ⟨a, b | aaa=aa, bbbb=aa⟩
Next:#15980 ⟨a, b | aaa=aa, bbbb=bb⟩

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. b10 ⇒ b7
  2. ab ⇒ b4
  3. a3 ⇒ a2 
# ab:aaa=aa,bbbb=ab b/a
bbbbbbbbbb=bbbbbbb
ab=bbbb
aaa=aa

Right Cayley graph

Left Cayley graph

Others with same cardinality

8 unique, 123 total

Σ#PresentationDescriptionRelated
91470⟨a, b | aaa=bb, abab=1⟩Finite non-Abelian group with 30 elements73 iso
91907⟨a, b | aab=a, bbbbb=1⟩Finite non-commutative monoid with 30 elements11 iso, 4 anti-iso
92443⟨a, b | aaa=1, abbbb=b⟩Finite non-commutative monoid with 30 elements14 iso, 5 anti-iso
106560⟨a, b | aaa=a, bbbb=ab⟩Finite non-commutative monoid with 30 elements
107013⟨a, b | ab=aa, bbbbb=b⟩Finite non-commutative monoid with 30 elements
1111058⟨a, b | abba=bbb, baba=1⟩Finite non-Abelian group with 30 elements8 iso
1113115⟨a, b | bbb=aaa, aaba=b⟩Finite non-commutative monoid with 30 elements
1120847⟨a, b | ab=aa, bbbbb=bb⟩Finite non-commutative monoid with 30 elements