#19706 ⟨a, b | aba=b, aaaaa=bb⟩

Up:Monoids with two generators and two relations
Prev:#19704 ⟨a, b | aba=b, aaaaa=aa⟩
Next:#19718 ⟨a, b | aba=b, aaabb=bb⟩

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. a15 ⇒ a5 [9]
2. ba10 ⇒ b [8]
3. ab ⇒ ba9 [7]
4. b2 ⇒ a5 [2]
# ab:aba=b,aaaaa=bb a/b
aaaaaaaaaaaaaaa=aaaaa
baaaaaaaaaa=b
ab=baaaaaaaaa
bb=aaaaa

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
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