#646 ⟨a, b | ba=ab, aaa=a⟩

Contents

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

Properties

• Sum of relation sides is 8
• Isomorphic to ℕ_{(3 = 1)} ⊕ ℕ
• 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⟩
• Cancellative quotient is isomorphic to ℤ₂ ⊕ ℕ
• Enveloping group is isomorphic to ℤ₂ ⊕ ℤ
• Group of units is isomorphic to ℤ₁
• 4 Archimedian components:
  • 1, a, b, ab

Staircase diagram

Rewriting system

1

# ab:ba=ab,aaa=a ab - -
ba=ab
aaa=a

Isomorphic instances

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