#658 ⟨a, b | ab=aa, ba=aa⟩

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 a has infinite order
• Not cancellative, because multiplication by a is not injective:
  • a ⋅ b = a² and a ⋅ a = a², however b ≠ a

Commutative structure

• Gröbner basis: ⟨a, b | ab=a²⟩
• 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:ab=aa,ba=aa ab - -
ab=aa
ba=aa

Isomorphic instances

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