#566 ⟨a, b | aaa=a, aba=b⟩

Contents

1. Properties
2. Commutative structure
3. Staircase diagram
4. Rewriting system
5. Isomorphic instances

Properties

• Sum of relation sides is 8
• Infinite non-cancellative commutative monoid
• Element b has infinite order
• Not cancellative, because multiplication by a is not injective:
  • a ⋅ a² = a and a ⋅ 1 = a, however a² ≠ 1

Commutative structure

• Gröbner basis: ⟨a, b | a³=a, a²b=b⟩
• Cancellative quotient is isomorphic to ℤ₂ ⊕ ℕ
• Enveloping group is isomorphic to ℤ₂ ⊕ ℤ
• Group of units is isomorphic to ℤ₁
• 3 Archimedian components:
  • 1, a, b

Staircase diagram

Rewriting system

1

# ab:aaa=a,aba=b ab - -
ba=ab
aaa=a
aab=b

Isomorphic instances

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