#2968 ⟨a, b | ab=a, baaaa=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 = a and a ⋅ 1 = a, however b ≠ 1
- Gröbner basis: ⟨a, b | ab=a, 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=a,baaaa=a ab - -
ab=a
ba=a
aaaa=a
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
23 total
| Σ | # | Presentation | Mapping |
|---|
| 10 | 8778 | ⟨a, b | ab=a, baaaab=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 8780 | ⟨a, b | ab=a, baaaba=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 8784 | ⟨a, b | ab=a, baabaa=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 8792 | ⟨a, b | ab=a, babaaa=a⟩ | φ(a) = a, φ(b) = b |
| 10 | 9049 | ⟨a, b | ab=a, baaaa=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 15577 | ⟨a, b | aab=aa, baaaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 15704 | ⟨a, b | aab=ba, abbbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 15872 | ⟨a, b | aba=ab, abbbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 24466 | ⟨a, b | ab=a, baaaabb=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24470 | ⟨a, b | ab=a, baaabab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24472 | ⟨a, b | ab=a, baaabba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24478 | ⟨a, b | ab=a, baabaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24480 | ⟨a, b | ab=a, baababa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24484 | ⟨a, b | ab=a, baabbaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24494 | ⟨a, b | ab=a, babaaab=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24496 | ⟨a, b | ab=a, babaaba=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24500 | ⟨a, b | ab=a, bababaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24508 | ⟨a, b | ab=a, babbaaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24993 | ⟨a, b | ab=a, baaaab=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 24997 | ⟨a, b | ab=a, baaaba=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 25005 | ⟨a, b | ab=a, baabaa=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 25021 | ⟨a, b | ab=a, babaaa=ab⟩ | φ(a) = a, φ(b) = b |
| 11 | 25535 | ⟨a, b | ab=a, baaaa=abb⟩ | φ(a) = a, φ(b) = b |