#2888 ⟨a, b | aa=a, babab=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 ⋅ a = a and a ⋅ 1 = a, however a ≠ 1
- Gröbner basis: ⟨a, b | a2=a, ab3=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:aa=a,babab=a ab - -
aa=a
ba=ab
abbb=a
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
28 total
| Σ | # | Presentation | Mapping |
|---|
| 10 | 5180 | ⟨a, b | aab=ba, babb=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5184 | ⟨a, b | aab=ba, bbab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 5264 | ⟨a, b | aba=ab, babb=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 8624 | ⟨a, b | aa=a, baabab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 8899 | ⟨a, b | aa=a, babab=aa⟩ | φ(a) = a, φ(b) = b |
| 11 | 12299 | ⟨a, b | aaab=bb, aaba=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12407 | ⟨a, b | aaba=bb, abaa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14608 | ⟨a, b | aaba=b, aaabb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14620 | ⟨a, b | aaba=b, abaab=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14622 | ⟨a, b | aaba=b, ababa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 14626 | ⟨a, b | aaba=b, abbaa=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15651 | ⟨a, b | aab=ab, babab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15687 | ⟨a, b | aab=ba, aabbb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15695 | ⟨a, b | aab=ba, ababb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15699 | ⟨a, b | aab=ba, abbab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15701 | ⟨a, b | aab=ba, abbba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15711 | ⟨a, b | aab=ba, baabb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15715 | ⟨a, b | aab=ba, babab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15717 | ⟨a, b | aab=ba, babba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15725 | ⟨a, b | aab=ba, bbaba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15729 | ⟨a, b | aab=ba, bbbaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15879 | ⟨a, b | aba=ab, baabb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15883 | ⟨a, b | aba=ab, babab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15885 | ⟨a, b | aba=ab, babba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24152 | ⟨a, b | aa=a, baaabab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24156 | ⟨a, b | aa=a, baabaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24695 | ⟨a, b | aa=a, baabab=aa⟩ | φ(a) = a, φ(b) = b |
| 11 | 25234 | ⟨a, b | aa=a, babab=aaa⟩ | φ(a) = a, φ(b) = b |