#4658 ⟨a, b | aaab=a, abba=b⟩

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 ⋅ a²b = a and a ⋅ 1 = a, however a²b ≠ 1

Commutative structure

• Gröbner basis: ⟨a, b | a³b=a, a²b²=b⟩
• Cancellative quotient is isomorphic to ℤ
• Enveloping group is isomorphic to ℤ
• Group of units is isomorphic to ℤ₁
• 2 Archimedian components:
  • 1, b

Staircase diagram

Rewriting system

1

# ab:aaab=a,abba=b ab - -
ba=ab
aaab=a
aabb=b

Isomorphic instances

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