#593 ⟨a, b | aab=b, aba=b⟩
Contents
- Properties
- Commutative structure
- Staircase diagram
- Rewriting system
- Isomorphic instances
- Sum of relation sides is 8
- Infinite non-cancellative commutative monoid
- Element a has infinite order
- Not cancellative, because multiplication by ab is not injective:
-
ab ⋅ a2 = ab and ab ⋅ 1 = ab, however a2 ≠ 1
- Gröbner basis: ⟨a, b | a2b=b⟩
- 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:aab=b,aba=b ab - -
ba=ab
aab=b
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
10 total
| Σ | # | Presentation | Mapping |
|---|
| 8 | 649 | ⟨a, b | ba=ab, aab=b⟩ | φ(a) = a, φ(b) = b |
| 8 | 651 | ⟨a, b | ba=ab, aba=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 6387 | ⟨a, b | aab=b, aaaba=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 6491 | ⟨a, b | aba=b, aaaba=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 6668 | ⟨a, b | aab=b, aaab=ba⟩ | φ(a) = a, φ(b) = b |
| 10 | 6671 | ⟨a, b | aab=b, aaba=ab⟩ | φ(a) = a, φ(b) = b |
| 10 | 6766 | ⟨a, b | aba=b, aaab=ab⟩ | φ(a) = a, φ(b) = b |
| 10 | 6771 | ⟨a, b | aba=b, aaba=ba⟩ | φ(a) = a, φ(b) = b |
| 10 | 6817 | ⟨a, b | aab=b, aba=aab⟩ | φ(a) = a, φ(b) = b |
| 10 | 6837 | ⟨a, b | aba=b, baa=aba⟩ | φ(a) = a, φ(b) = b |