#2123 ⟨a, b | aba=b, bbbb=b⟩
Contents
- Properties
- Commutative structure
- Staircase diagram
- Rewriting system
- Isomorphic instances
- Sum of relation sides is 9
- Infinite non-cancellative commutative monoid
- Element a has infinite order
- Not cancellative, because multiplication by b is not injective:
-
b ⋅ b3 = b and b ⋅ 1 = b, however b3 ≠ 1
- Gröbner basis: ⟨a, b | a2b=b, b4=b⟩
- 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:aba=b,bbbb=b ab - -
ba=ab
aab=b
bbbb=b
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
17 total
| Σ | # | Presentation | Mapping |
|---|
| 11 | 12605 | ⟨a, b | abba=bb, bbbb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 19189 | ⟨a, b | aba=b, aabbbb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 19201 | ⟨a, b | aba=b, ababbb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 19205 | ⟨a, b | aba=b, abbabb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 19207 | ⟨a, b | aba=b, abbbab=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 19209 | ⟨a, b | aba=b, abbbba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 19219 | ⟨a, b | aba=b, baabbb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 19221 | ⟨a, b | aba=b, bababb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 19223 | ⟨a, b | aba=b, babbab=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 19227 | ⟨a, b | aba=b, bbaabb=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 19753 | ⟨a, b | aba=b, abbbb=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19754 | ⟨a, b | aba=b, abbbb=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19767 | ⟨a, b | aba=b, babbb=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 19768 | ⟨a, b | aba=b, babbb=ba⟩ | φ(a) = a, φ(b) = b |
| 11 | 19771 | ⟨a, b | aba=b, bbabb=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 20315 | ⟨a, b | aba=b, bbbb=aab⟩ | φ(a) = a, φ(b) = b |
| 11 | 20316 | ⟨a, b | aba=b, bbbb=aba⟩ | φ(a) = a, φ(b) = b |