#14357 ⟨a, b | aaaa=a, baaab=a⟩

Contents

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

Properties

• Sum of relation sides is 11
• Infinite non-cancellative commutative monoid
• Element b has infinite order
• Not cancellative, because multiplication by a is not injective:
  • a ⋅ a³ = a and a ⋅ 1 = a, however a³ ≠ 1

Commutative structure

• Gröbner basis: ⟨a, b | ab²=a², a⁴=a⟩
• Cancellative quotient is isomorphic to ℤ₆
• Enveloping group is isomorphic to ℤ₆
• Group of units is isomorphic to ℤ₁
• 3 Archimedian components:
  • 1, a, b

Staircase diagram

Rewriting system

1

# ab:aaaa=a,baaab=a ab - -
ba=ab
abb=aa
aaaa=a

Isomorphic instances

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