#20248 ⟨a, b | aba=b, aaaa=aaa⟩

Contents

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

Properties

• Sum of relation sides is 11
• Infinite non-cancellative commutative monoid
• Element b has infinite order
• Not cancellative, because multiplication by b is not injective:
  • b ⋅ a = b and b ⋅ 1 = b, however a ≠ 1

Commutative structure

• Gröbner basis: ⟨a, b | ab=b, a⁴=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:aba=b,aaaa=aaa ab - -
ab=b
ba=b
aaaa=aaa