#3193 ⟨a, b | ab=a, bbb=aaa⟩
Contents
- Properties
- Commutative structure
- Finite structure
- Staircase diagram
- Cayley table
- Right Cayley graph
- Rewriting system
- Same cardinality
- Isomorphic instances
- Sum of relation sides is 9
- Finite commutative monoid with 6 elements
- 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, b3=a3, a4=a⟩
- Cancellative quotient is isomorphic to ℤ3
- Enveloping group is isomorphic to ℤ3
- Group of units is isomorphic to ℤ1
- 2 Archimedian components:
An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.
- 1 non-trivial idempotent:
- Order of generators:
- a: index 1, period 3
- b: index 3, period 1
- Histogram:
| index 1, period 1 | 1 element | a3 |
| index 1, period 3 | 2 elements | a, a2 |
| index 2, period 1 | 1 element | b2 |
| index 3, period 1 | 1 element | b |
Idempotents are shown in bold.
|
1 | a | b | a2 | b2 | a3 |
|---|
| 1 | 1 | a | b | a2 | b2 | a3 |
|---|
| a | a | a2 | a | a3 | a | a |
|---|
| b | b | a | b2 | a2 | a3 | a3 |
|---|
| a2 | a2 | a3 | a2 | a | a2 | a2 |
|---|
| b2 | b2 | a | a3 | a2 | a3 | a3 |
|---|
| a3 | a3 | a | a3 | a2 | a3 | a3 |
|---|
Idempotents are shown in bold.
- Reduction order:
- Left-to-right shortlex with a < b
- Certificate: derivations of all rewriting rules from the defining relations.
# ab:ab=a,bbb=aaa ab - -
ab=a
ba=a
bbb=aaa
aaaa=a
19 unique, 1916 total
| Σ | # | Presentation | Properties | Description | φ |
|---|
| 6 | 22 | ⟨a, b | ab=aa, bb=1⟩ | Noc 6 | | 107 |
| 6 | 26 | ⟨a, b | aaa=1, abb=1⟩ | Grp Com 6 | ℤ6 | 1373 |
| 7 | 160 | ⟨a, b | ab=aa, bb=b⟩ | Noc 6 | | 4 |
| 7 | 245 | ⟨a, b | aa=a, bbb=a⟩ | Noc Com 6 | ℕ(6 = 3) | 49 |
| 7 | 257 | ⟨a, b | aa=b, bbb=a⟩ | Noc Com 6 | ℕ(6 = 1) | 61 |
| 7 | 258 | ⟨a, b | aa=b, bbb=b⟩ | Noc Com 6 | ℕ(6 = 2) | 55 |
| 8 | 639 | ⟨a, b | ab=aa, baa=b⟩ | Noc 6 | | 13 |
| 8 | 644 | ⟨a, b | ab=aa, bbb=a⟩ | Noc Com 6 | ℕ(6 = 4) | 46 |
| 8 | 893 | ⟨a, b | aa=a, abbb=b⟩ | Noc 6 | | 19 |
| 8 | 1011 | ⟨a, b | ab=a, baa=bb⟩ | Noc 6 | | 13 |
| 9 | 1427 | ⟨a, b | abba=b, baba=1⟩ | Grp 6 | | 66 |
| 9 | 1581 | ⟨a, b | aaa=aa, abb=b⟩ | Noc 6 | | 5 |
| 9 | 1648 | ⟨a, b | aab=bb, abb=a⟩ | Noc Com 6 | | 13 |
| 9 | 1686 | ⟨a, b | abb=aa, bbb=a⟩ | Noc Com 6 | ℕ(6 = 5) | 49 |
| 9 | 3075 | ⟨a, b | ab=a, aaaa=bb⟩ | Noc Com 6 | | 14 |
| 9 | 3132 | ⟨a, b | ab=a, bbbb=aa⟩ | Noc Com 6 | | 5 |
| 9 | 3134 | ⟨a, b | ab=a, bbbb=ba⟩ | Noc 6 | | 2 |
| 9 | 3199 | ⟨a, b | ab=a, bbb=bba⟩ | Noc 6 | | 3 |
| 11 | 24145 | ⟨a, b | aa=a, abbbbba=b⟩ | Noc Com 6 | | |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
9 total
| Σ | # | Presentation | Mapping |
|---|
| 10 | 9271 | ⟨a, b | ab=a, aaab=bbb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9279 | ⟨a, b | ab=a, aaba=bbb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9295 | ⟨a, b | ab=a, abaa=bbb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25435 | ⟨a, b | ab=a, aaabb=bbb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25451 | ⟨a, b | ab=a, aabab=bbb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25459 | ⟨a, b | ab=a, aabba=bbb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25483 | ⟨a, b | ab=a, abaab=bbb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25491 | ⟨a, b | ab=a, ababa=bbb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25507 | ⟨a, b | ab=a, abbaa=bbb⟩ | φ(a) = a, φ(b) = b |