#14981 ⟨a, b | aaa=bb, aabaab=1⟩

Up:Monoids with two generators and two relations
Prev:#14908 ⟨a, b | aaa=aa, bbbbbb=1⟩
Next:#14993 ⟨a, b | aaa=bb, ababbb=1⟩

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.
The order of x is the least n such that xn = 1.

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. a21 ⇒ 1 [6]
2. ab ⇒ ba8 [5]
3. b2 ⇒ a3 [1]
# ab:aaa=bb,aabaab=1 a/b
aaaaaaaaaaaaaaaaaaaaa=1
ab=baaaaaaaa
bb=aaa

Same cardinality

2 unique, 5 total

Σ#PresentationDescriptionRelated
106101⟨a, b | aab=a, bbbbbb=1⟩Finite non-commutative monoid with 42 elements3 iso
1120647⟨a, b | ab=aa, bbbbbb=b⟩Finite non-commutative monoid with 42 elements

Isomorphic instances

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

7 total

Σ#PresentationMapping
1114989⟨a, b | aaa=bb, abaaba=1⟩φ(a) = a, φ(b) = b
1126093⟨a, b | aa=1, abbabbbbb=1⟩φ(a) = baa, φ(b) = a
1126108⟨a, b | aa=1, abbbbbabb=1⟩φ(a) = baa, φ(b) = a
1126158⟨a, b | aa=1, babbabbbb=1⟩φ(a) = baa, φ(b) = a
1126162⟨a, b | aa=1, babbbbbab=1⟩φ(a) = baa, φ(b) = a
1126175⟨a, b | aa=1, bbabbabbb=1⟩φ(a) = baa, φ(b) = a
1126704⟨a, b | aa=1, bbabbbbb=a⟩φ(a) = baa, φ(b) = a