#588 ⟨a, b | aab=a, bab=a⟩
Contents
- Properties
- Commutative structure
- Staircase diagram
- Rewriting system
- Isomorphic instances
- Sum of relation sides is 8
- Infinite non-cancellative commutative monoid
- Element b has infinite order
- Not cancellative, because multiplication by a is not injective:
-
a ⋅ a2 = a and a ⋅ 1 = a, however a2 ≠ 1
- Gröbner basis: ⟨a, b | ab=a2, a3=a⟩
- Cancellative quotient is isomorphic to ℤ2
- Enveloping group is isomorphic to ℤ2
- 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=a,bab=a ab - -
ab=aa
ba=aa
aaa=a
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
41 total
| Σ | # | Presentation | Mapping |
|---|
| 8 | 599 | ⟨a, b | aba=a, bab=a⟩ | φ(a) = a, φ(b) = b |
| 8 | 638 | ⟨a, b | ab=aa, baa=a⟩ | φ(a) = a, φ(b) = b |
| 8 | 640 | ⟨a, b | ab=aa, bab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 4124 | ⟨a, b | aab=aaa, baa=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 4160 | ⟨a, b | abb=aab, bab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 4166 | ⟨a, b | abb=aba, bab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 4178 | ⟨a, b | baa=abb, bab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 4181 | ⟨a, b | baa=abb, bba=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 4191 | ⟨a, b | bab=aba, bba=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 6350 | ⟨a, b | aab=a, baaaa=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 6356 | ⟨a, b | aab=a, baabb=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 6360 | ⟨a, b | aab=a, babab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 6362 | ⟨a, b | aab=a, babba=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 6478 | ⟨a, b | aba=a, babab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 6493 | ⟨a, b | aba=b, aaabb=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 6497 | ⟨a, b | aba=b, aabab=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 6499 | ⟨a, b | aba=b, aabba=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 6503 | ⟨a, b | aba=b, abaab=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 6505 | ⟨a, b | aba=b, ababa=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 6515 | ⟨a, b | aba=b, baaab=b⟩ | φ(a) = b, φ(b) = a |
| 10 | 6604 | ⟨a, b | aab=a, aaab=ba⟩ | φ(a) = a, φ(b) = b |
| 10 | 6608 | ⟨a, b | aab=a, aaba=ba⟩ | φ(a) = a, φ(b) = b |
| 10 | 6616 | ⟨a, b | aab=a, abaa=ba⟩ | φ(a) = a, φ(b) = b |
| 10 | 6628 | ⟨a, b | aab=a, abbb=ba⟩ | φ(a) = a, φ(b) = b |
| 10 | 6631 | ⟨a, b | aab=a, baaa=ab⟩ | φ(a) = a, φ(b) = b |
| 10 | 6634 | ⟨a, b | aab=a, baab=aa⟩ | φ(a) = a, φ(b) = b |
| 10 | 6638 | ⟨a, b | aab=a, baba=aa⟩ | φ(a) = a, φ(b) = b |
| 10 | 6643 | ⟨a, b | aab=a, babb=ab⟩ | φ(a) = a, φ(b) = b |
| 10 | 6731 | ⟨a, b | aba=a, aaab=ba⟩ | φ(a) = a, φ(b) = b |
| 10 | 6768 | ⟨a, b | aba=b, aaab=bb⟩ | φ(a) = b, φ(b) = a |
| 10 | 6772 | ⟨a, b | aba=b, aaba=bb⟩ | φ(a) = b, φ(b) = a |
| 10 | 6774 | ⟨a, b | aba=b, aabb=ab⟩ | φ(a) = b, φ(b) = a |
| 10 | 6775 | ⟨a, b | aba=b, aabb=ba⟩ | φ(a) = b, φ(b) = a |
| 10 | 6778 | ⟨a, b | aba=b, abab=ab⟩ | φ(a) = b, φ(b) = a |
| 10 | 6779 | ⟨a, b | aba=b, abab=ba⟩ | φ(a) = b, φ(b) = a |
| 10 | 6782 | ⟨a, b | aba=b, abba=ab⟩ | φ(a) = b, φ(b) = a |
| 10 | 6789 | ⟨a, b | aba=b, baab=ab⟩ | φ(a) = b, φ(b) = a |
| 10 | 6813 | ⟨a, b | aab=a, bab=aab⟩ | φ(a) = a, φ(b) = b |
| 10 | 6831 | ⟨a, b | aba=a, bab=aba⟩ | φ(a) = a, φ(b) = b |
| 10 | 6835 | ⟨a, b | aba=b, abb=aba⟩ | φ(a) = b, φ(b) = a |
| 10 | 6839 | ⟨a, b | aba=b, bab=aba⟩ | φ(a) = b, φ(b) = a |