#891 ⟨a, b | aa=a, abba=b⟩

Up:Monoids with two generators and two relations
Prev:#890 ⟨a, b | aa=a, abba=a⟩
Next:#892 ⟨a, b | aa=a, abbb=a⟩

Quick links

  1. Properties
  2. Elements
  3. Staircase diagram
  4. Cayley table
  5. Right Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Elements

The zero element z satisfies zy = yz = z for all y.
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.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1ab
11ab
aaab
bbbb

Right Cayley graph

Idempotents are shown in bold.

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. a2 ⇒ a [1]
2. ab ⇒ b [3]
3. ba ⇒ b [6]
4. b2 ⇒ b [7]
# ab:aa=a,abba=b ab
aa=a
ab=b
ba=b
bb=b

Same cardinality

4 unique, 2195 total

Σ#PresentationDescriptionRelated
56⟨a, b | aa=b, ab=1⟩Isomorphic to ℤ32029 iso
659⟨a, b | aa=b, ab=a⟩Isomorphic to ℕ(3 = 1)61 iso
660⟨a, b | aa=b, ab=b⟩Isomorphic to ℕ(3 = 2)23 iso
663⟨a, b | ab=a, ba=b⟩Finite non-commutative monoid with 3 elements61 iso, 17 anti-iso

Isomorphic instances

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

12 total

Σ#PresentationMapping
92869⟨a, b | aa=a, aabba=b⟩φ(a) = a, φ(b) = b
92875⟨a, b | aa=a, ababa=b⟩φ(a) = a, φ(b) = b
108581⟨a, b | aa=a, aaabba=b⟩φ(a) = a, φ(b) = b
108587⟨a, b | aa=a, aababa=b⟩φ(a) = a, φ(b) = b
108591⟨a, b | aa=a, aabbaa=b⟩φ(a) = a, φ(b) = b
108601⟨a, b | aa=a, abaaba=b⟩φ(a) = a, φ(b) = b
1124057⟨a, b | aa=a, aaaabba=b⟩φ(a) = a, φ(b) = b
1124065⟨a, b | aa=a, aaababa=b⟩φ(a) = a, φ(b) = b
1124069⟨a, b | aa=a, aaabbaa=b⟩φ(a) = a, φ(b) = b
1124079⟨a, b | aa=a, aabaaba=b⟩φ(a) = a, φ(b) = b
1124083⟨a, b | aa=a, aababaa=b⟩φ(a) = a, φ(b) = b
1124107⟨a, b | aa=a, abaaaba=b⟩φ(a) = a, φ(b) = b