#2192 ⟨a, b | ab=aa, baaa=a⟩
Contents
- Properties
- Commutative structure
- Staircase diagram
- Rewriting system
- Isomorphic instances
- Sum of relation sides is 9
- Infinite non-cancellative commutative monoid
- Element b has infinite order
- Not cancellative, because multiplication by a is not injective:
-
a ⋅ b = a2 and a ⋅ a = a2, however b ≠ a
- Gröbner basis: ⟨a, b | ab=a2, a4=a⟩
- Cancellative quotient is isomorphic to ℤ3
- Enveloping group is isomorphic to ℤ3
- 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:ab=aa,baaa=a ab - -
ab=aa
ba=aa
aaaa=a
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
28 total
| Σ | # | Presentation | Mapping |
|---|
| 9 | 2194 | ⟨a, b | ab=aa, baab=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 2196 | ⟨a, b | ab=aa, baba=a⟩ | φ(a) = a, φ(b) = b |
| 9 | 2198 | ⟨a, b | ab=aa, babb=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 4663 | ⟨a, b | aaab=a, baab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 4665 | ⟨a, b | aaab=a, baba=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 4706 | ⟨a, b | aaba=a, baab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 4720 | ⟨a, b | aaba=b, abab=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 4722 | ⟨a, b | aaba=b, abba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12896 | ⟨a, b | aab=aaa, baaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12898 | ⟨a, b | aab=aaa, baab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12900 | ⟨a, b | aab=aaa, baba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 12926 | ⟨a, b | aba=aaa, baab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 13001 | ⟨a, b | abb=aab, aaba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 13007 | ⟨a, b | abb=aab, abba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 13021 | ⟨a, b | abb=aba, abaa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 13023 | ⟨a, b | abb=aba, abab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 13025 | ⟨a, b | abb=aba, abba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 13027 | ⟨a, b | abb=aba, abbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 13030 | ⟨a, b | abb=aba, baab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 13066 | ⟨a, b | baa=abb, aaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 13069 | ⟨a, b | baa=abb, aaba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 13074 | ⟨a, b | baa=abb, abab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 13102 | ⟨a, b | bab=aba, aaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 13104 | ⟨a, b | bab=aba, aaba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 13105 | ⟨a, b | bab=aba, aaba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 13106 | ⟨a, b | bab=aba, aabb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 13107 | ⟨a, b | bab=aba, abab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 13108 | ⟨a, b | bab=aba, abba=a⟩ | φ(a) = a, φ(b) = b |