#14 ⟨a, b | abba=1⟩
Quick links
- Properties
- Rewriting system
- Other submonoids of same group
- Isomorphic instances
- Presentation has sum-of-sides 4
- Infinite non-Abelian group
- Inverses of generators:
- a-1 = ad
- b-1 = bc
- c-1 = d
- d-1 = c
- Reduced group presentation: ⟨a, b | aabb⟩
- Isomorphism: φ(a) = b, φ(b) = a
- Auxiliary generators:
- c = aa
- d = bb
- Reduction order:
- Left-to-right shortlex with a < b < c < d
- Certificate: derivations of all rewriting rules from the defining relations.
# ab:abba=1 abcd aa=c,bb=d magic:0
aa=c
bb=d
ca=ac
cb=bc
cd=1
da=ad
db=bd
dc=1
2 unique, 2 total
| Σ | # | Presentation | Description | Related |
|---|
| 4 | 20 | ⟨a, b | aba=b⟩ | Infinite cancellative non-commutative monoid | |
| 4 | 23 | ⟨a, b | bb=aa⟩ | Infinite cancellative non-commutative monoid | |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
14 total
| Σ | # | Presentation | Mapping |
|---|
| 6 | 63 | ⟨a, b | aababa=1⟩ | φ(a) = a, φ(b) = adb |
| 6 | 68 | ⟨a, b | abaaab=1⟩ | φ(a) = b, φ(b) = bca |
| 8 | 286 | ⟨a, b | aaabaaba=1⟩ | φ(a) = b, φ(b) = bcabc |
| 8 | 299 | ⟨a, b | aabaaaab=1⟩ | φ(a) = b, φ(b) = bcbca |
| 8 | 302 | ⟨a, b | aabaabaa=1⟩ | φ(a) = a, φ(b) = adbad |
| 8 | 322 | ⟨a, b | abaaaaba=1⟩ | φ(a) = b, φ(b) = bcabc |
| 10 | 1308 | ⟨a, b | aaaabaaaba=1⟩ | φ(a) = b, φ(b) = bcbcabc |
| 10 | 1337 | ⟨a, b | aaabaaaaab=1⟩ | φ(a) = b, φ(b) = bcbcbca |
| 10 | 1340 | ⟨a, b | aaabaaabaa=1⟩ | φ(a) = b, φ(b) = bcabcbc |
| 10 | 1392 | ⟨a, b | aabaaaaaba=1⟩ | φ(a) = b, φ(b) = bcbcabc |
| 10 | 1420 | ⟨a, b | aabababaab=1⟩ | φ(a) = bca, φ(b) = adbb |
| 10 | 1492 | ⟨a, b | abaabaabab=1⟩ | φ(a) = adb, φ(b) = bcaa |
| 10 | 1495 | ⟨a, b | abaabababa=1⟩ | φ(a) = adb, φ(b) = abca |
| 10 | 1508 | ⟨a, b | ababaabaab=1⟩ | φ(a) = adb, φ(b) = bcaa |