#7129 ⟨a, b | ba=ab, aabb=aa⟩
Contents
- Properties
- Commutative structure
- Staircase diagram
- Rewriting system
- Isomorphic instances
- Sum of relation sides is 10
- Infinite non-cancellative commutative monoid
- Element a has infinite order
- Not cancellative, because multiplication by a2 is not injective:
-
a2 ⋅ b2 = a2 and a2 ⋅ 1 = a2, however b2 ≠ 1
- Gröbner basis: ⟨a, b | a2b2=a2⟩
- Cancellative quotient is isomorphic to ℤ2 ⊕ ℕ
- Enveloping group is isomorphic to ℤ2 ⊕ ℤ
- Group of units is isomorphic to ℤ1
- 3 Archimedian components:
- Reduction order:
- Left-to-right shortlex with a < b
- Certificate: derivations of all rewriting rules from the defining relations.
# ab:ba=ab,aabb=aa ab - -
ba=ab
aabb=aa
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
3 total
| Σ | # | Presentation | Mapping |
|---|
| 10 | 7132 | ⟨a, b | ba=ab, abab=aa⟩ | φ(a) = a, φ(b) = b |
| 10 | 7135 | ⟨a, b | ba=ab, abba=aa⟩ | φ(a) = a, φ(b) = b |
| 10 | 7137 | ⟨a, b | ba=ab, abba=bb⟩ | φ(a) = b, φ(b) = a |