#2214 ⟨a, b | ba=ab, aabb=a⟩
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 a is not injective:
-
a ⋅ ab2 = a and a ⋅ 1 = a, however ab2 ≠ 1
- Gröbner basis: ⟨a, b | a2b2=a⟩
- Cancellative quotient is isomorphic to ℤ
- 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:ba=ab,aabb=a ab - -
ba=ab
aabb=a
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
15 total
| Σ | # | Presentation | Mapping |
|---|
| 9 | 2215 | ⟨a, b | ba=ab, abab=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 2216 | ⟨a, b | ba=ab, abba=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 2217 | ⟨a, b | ba=ab, abba=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 4735 | ⟨a, b | aabb=a, baab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 4743 | ⟨a, b | abab=a, baab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 4745 | ⟨a, b | abab=a, baba=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 4747 | ⟨a, b | abba=a, baab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12939 | ⟨a, b | aba=aab, aabb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12943 | ⟨a, b | aba=aab, abab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12945 | ⟨a, b | aba=aab, abba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12950 | ⟨a, b | aba=aab, baab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12952 | ⟨a, b | aba=aab, baba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 13050 | ⟨a, b | baa=aab, aabb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 13052 | ⟨a, b | baa=aab, abab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 13058 | ⟨a, b | baa=aab, baab=a⟩ | φ(a) = a, φ(b) = b |