#891 ⟨a, b | aa=a, abba=b⟩

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Contents

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

Properties

Commutative structure

Finite structure

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:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 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#PresentationPropertiesDescriptionφ
56⟨a, b | aa=b, ab=1⟩Grp Com 3ℤ32029
659⟨a, b | aa=b, ab=a⟩Noc Com 3ℕ(3 = 1)61
660⟨a, b | aa=b, ab=b⟩Noc Com 3ℕ(3 = 2)23
663⟨a, b | ab=a, ba=b⟩Noc 378

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