#7186 ⟨a, b | ba=ab, aab=aaa⟩

Contents

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

Properties

• Sum of relation sides is 10
• Infinite non-cancellative commutative monoid
• Element a has infinite order
• Not cancellative, because multiplication by a is not injective:
  • a ⋅ ab = a³ and a ⋅ a² = a³, however ab ≠ a²

Commutative structure

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

Staircase diagram

Rewriting system

1

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

Isomorphic instances

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