#162 ⟨a, b | ba=ab, bb=b⟩
Contents
- Properties
- Commutative structure
- Staircase diagram
- Rewriting system
- Isomorphic instances
- Sum of relation sides is 7
- Isomorphic to ℕ(2 = 1) ⊕ ℕ
- Infinite non-cancellative commutative monoid
- Element a has infinite order
- Not cancellative, because multiplication by b is not injective:
-
b ⋅ b = b and b ⋅ 1 = b, however b ≠ 1
- Gröbner basis: ⟨a, b | b2=b⟩
- Cancellative quotient is isomorphic to ℕ
- Enveloping group is isomorphic to ℤ
- Group of units is isomorphic to ℤ1
- 4 Archimedian components:
- Reduction order:
- Left-to-right shortlex with a < b
- Certificate: derivations of all rewriting rules from the defining relations.
# ab:ba=ab,bb=b ab - -
ba=ab
bb=b
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
8 total
| Σ | # | Presentation | Mapping |
|---|
| 8 | 957 | ⟨a, b | aa=a, aab=ba⟩ | φ(a) = b, φ(b) = a |
| 9 | 3005 | ⟨a, b | aa=a, aaab=ba⟩ | φ(a) = b, φ(b) = a |
| 9 | 3142 | ⟨a, b | aa=a, baa=aab⟩ | φ(a) = b, φ(b) = a |
| 10 | 8845 | ⟨a, b | aa=a, aaaab=ba⟩ | φ(a) = b, φ(b) = a |
| 10 | 9122 | ⟨a, b | aa=a, aaab=baa⟩ | φ(a) = b, φ(b) = a |
| 11 | 24593 | ⟨a, b | aa=a, aaaaab=ba⟩ | φ(a) = b, φ(b) = a |
| 11 | 25126 | ⟨a, b | aa=a, aaaab=baa⟩ | φ(a) = b, φ(b) = a |
| 11 | 25686 | ⟨a, b | aa=a, baaa=aaab⟩ | φ(a) = b, φ(b) = a |