#5064 ⟨a, b | aaa=ab, babb=a⟩
Contents
- Properties
- Commutative structure
- Staircase diagram
- Rewriting system
- Isomorphic instances
- Sum of relation sides is 10
- Infinite non-cancellative commutative monoid
- Element b has infinite order
- Not cancellative, because multiplication by a is not injective:
-
a ⋅ b3 = a and a ⋅ 1 = a, however b3 ≠ 1
- Gröbner basis: ⟨a, b | a3=ab, ab3=a⟩
- Cancellative quotient is isomorphic to ℤ6
- Enveloping group is isomorphic to ℤ6
- 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:aaa=ab,babb=a ab - -
ba=ab
aaa=ab
abbb=a
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
6 total
| Σ | # | Presentation | Mapping |
|---|
| 11 | 14624 | ⟨a, b | aaba=b, ababb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14628 | ⟨a, b | aaba=b, abbab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14630 | ⟨a, b | aaba=b, abbba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15475 | ⟨a, b | aaa=ab, baaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15477 | ⟨a, b | aaa=ab, baaba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15481 | ⟨a, b | aaa=ab, babaa=a⟩ | φ(a) = a, φ(b) = b |