#21070 ⟨a, b | ba=ab, aabb=aaa⟩
Contents
- Properties
- Commutative structure
- Staircase diagram
- Rewriting system
- Isomorphic instances
- Sum of relation sides is 11
- Infinite non-cancellative commutative monoid
- Element a has infinite order
- Not cancellative, because multiplication by a is not injective:
-
a ⋅ ab2 = a3 and a ⋅ a2 = a3, however ab2 ≠ a2
- Gröbner basis: ⟨a, b | a3=a2b2⟩
- Cancellative quotient is isomorphic to ℕ
- Enveloping group is isomorphic to 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=aaa ab - -
ba=ab
aabb=aaa
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
3 total
| Σ | # | Presentation | Mapping |
|---|
| 11 | 21074 | ⟨a, b | ba=ab, abab=aaa⟩ | φ(a) = a, φ(b) = b |
| 11 | 21078 | ⟨a, b | ba=ab, abba=aaa⟩ | φ(a) = a, φ(b) = b |
| 11 | 21083 | ⟨a, b | ba=ab, abba=bbb⟩ | φ(a) = b, φ(b) = a |