#1685 ⟨a, b | aba=bb, bbb=b⟩
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- Properties
- Staircase diagram
- Rewriting system
- Isomorphic instances
- Presentation has sum-of-sides 9
- Infinite non-cancellative commutative monoid
- Not cancellative, because multiplication by ab is not injective:
-
ab ⋅ b2 = ab and ab ⋅ 1 = ab, however b2 ≠ 1
- Commutative Gröbner basis: ⟨a, b | b3=b, a2b=b2⟩
- Cancellative quotient is isomorphic to ℤ4
- Enveloping group is isomorphic to ℤ4
- 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:aba=bb,bbb=b ab
ba=ab
aab=bb
bbb=b
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
25 total
| Σ | # | Presentation | Mapping |
|---|
| 9 | 1994 | ⟨a, b | aaa=a, baab=a⟩ | φ(a) = b, φ(b) = a |
| 10 | 5203 | ⟨a, b | aab=bb, abba=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 5281 | ⟨a, b | aba=bb, aabb=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 5283 | ⟨a, b | aba=bb, abab=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 5289 | ⟨a, b | aba=bb, baab=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 12160 | ⟨a, b | aaaa=aa, baab=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 12256 | ⟨a, b | aaab=ab, baab=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 12272 | ⟨a, b | aaab=ba, aabb=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 12276 | ⟨a, b | aaab=ba, abab=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 12278 | ⟨a, b | aaab=ba, abba=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 12284 | ⟨a, b | aaab=ba, baab=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 12286 | ⟨a, b | aaab=ba, baba=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 12290 | ⟨a, b | aaab=ba, bbaa=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 12366 | ⟨a, b | aaba=ab, baba=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 12382 | ⟨a, b | aaba=ba, abab=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 12390 | ⟨a, b | aaba=ba, baab=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 14711 | ⟨a, b | aabb=a, babbb=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 14775 | ⟨a, b | abab=a, babbb=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 14838 | ⟨a, b | abba=b, aaaba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 15742 | ⟨a, b | aab=bb, aaaba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 15910 | ⟨a, b | aba=bb, aaaba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 18740 | ⟨a, b | aaa=a, baaaab=a⟩ | φ(a) = b, φ(b) = a |
| 11 | 19284 | ⟨a, b | aaa=a, baaab=aa⟩ | φ(a) = b, φ(b) = a |
| 11 | 19780 | ⟨a, b | aaa=a, aaaa=bab⟩ | φ(a) = b, φ(b) = a |
| 11 | 19828 | ⟨a, b | aaa=a, baab=aaa⟩ | φ(a) = b, φ(b) = a |