#240 ⟨a, b | aa=a, aba=b⟩
Contents
- Properties
- Commutative structure
- Staircase diagram
- Rewriting system
- Isomorphic instances
- Sum of relation sides is 7
- Infinite non-cancellative commutative monoid
- Element b has infinite order
- Not cancellative, because multiplication by a is not injective:
-
a ⋅ a = a and a ⋅ 1 = a, however a ≠ 1
- Gröbner basis: ⟨a, b | a2=a, ab=b⟩
- 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:aa=a,aba=b ab - -
aa=a
ab=b
ba=b
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
8 total
| Σ | # | Presentation | Mapping |
|---|
| 8 | 885 | ⟨a, b | aa=a, aaba=b⟩ | φ(a) = a, φ(b) = b |
| 9 | 2861 | ⟨a, b | aa=a, aaaba=b⟩ | φ(a) = a, φ(b) = b |
| 9 | 2865 | ⟨a, b | aa=a, aabaa=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 8573 | ⟨a, b | aa=a, aaaaba=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 8577 | ⟨a, b | aa=a, aaabaa=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24049 | ⟨a, b | aa=a, aaaaaba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24053 | ⟨a, b | aa=a, aaaabaa=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24061 | ⟨a, b | aa=a, aaabaaa=b⟩ | φ(a) = a, φ(b) = b |