#6381 ⟨a, b | aab=a, bbbbb=b⟩

Up:Monoids with two generators and two relations
Prev:#6380 ⟨a, b | aab=a, bbbbb=a⟩
Next:#6382 ⟨a, b | aab=b, aaaaa=a⟩

Quick links

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

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. a5 ⇒ a [7]
2. ab ⇒ a4 [6]
3. b5 ⇒ b [2]
# ab:aab=a,bbbbb=b a/b
aaaaa=a
ab=aaaa
bbbbb=b

Same cardinality

7 unique, 12 total

Σ#PresentationDescriptionRelated
104749⟨a, b | abba=b, baab=a⟩Finite non-commutative monoid with 25 elements
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
1119706⟨a, b | aba=b, aaaaa=bb⟩Finite non-commutative monoid with 25 elements

Isomorphic instances

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

1 total

Σ#PresentationMapping
1114792⟨a, b | abab=a, bbbbb=b⟩φ(a) = aa, φ(b) = b