#891 ⟨a, b | aa=a, abba=b⟩

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Properties

Element profile

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. a2 ⇒ a
  2. ab ⇒ b
  3. ba ⇒ b
  4. b2 ⇒ b 
# ab:aa=a,abba=b ab
aa=a
ab=b
ba=b
bb=b

Staircase diagram

Cayley table

Idempotents are shown in bold.

1ab
11ab
aaab
bbbb

Right Cayley graph

Idempotents are shown in bold.

Others with 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

Other 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