#1601 ⟨a, b | aaa=bb, aab=b⟩
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 7 elements
- Not cancellative, because multiplication by ab is not injective:
-
ab ⋅ b2 = b and ab ⋅ a = b, however b2 ≠ a
- Gröbner basis: ⟨a, b | b3=ab, a2b=b, a3=b2⟩
- Cancellative quotient is isomorphic to ℤ4
- Enveloping group is isomorphic to ℤ4
- 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 3, period 2
- b: index 1, period 4
- Histogram:
| index 1, period 1 | 1 element | ab2 |
| index 1, period 2 | 1 element | b2 |
| index 1, period 4 | 2 elements | b, ab |
| index 2, period 1 | 1 element | a2 |
| index 3, period 2 | 1 element | a |
Idempotents are shown in bold.
|
1 | a | b | a2 | ab | b2 | ab2 |
|---|
| 1 | 1 | a | b | a2 | ab | b2 | ab2 |
|---|
| a | a | a2 | ab | b2 | b | ab2 | b2 |
|---|
| b | b | ab | b2 | b | ab2 | ab | b |
|---|
| a2 | a2 | b2 | b | ab2 | ab | b2 | ab2 |
|---|
| ab | ab | b | ab2 | ab | b2 | b | ab |
|---|
| b2 | b2 | ab2 | ab | b2 | b | ab2 | b2 |
|---|
| ab2 | ab2 | b2 | b | ab2 | ab | b2 | ab2 |
|---|
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:aaa=bb,aab=b ab - -
ba=ab
aaa=bb
aab=b
bbb=ab
22 unique, 1218 total
| Σ | # | Presentation | Properties | Description | φ |
|---|
| 7 | 116 | ⟨a, b | aaa=b, abb=1⟩ | Grp Com 7 | ℤ7 | 763 |
| 8 | 577 | ⟨a, b | aaa=b, abb=a⟩ | Noc Com 7 | ℕ(7 = 1) | 59 |
| 8 | 578 | ⟨a, b | aaa=b, abb=b⟩ | Noc Com 7 | ℕ(7 = 3) | 44 |
| 8 | 913 | ⟨a, b | aa=b, abbb=b⟩ | Noc Com 7 | ℕ(7 = 2) | 82 |
| 8 | 937 | ⟨a, b | ab=a, baaa=b⟩ | Noc 7 | | 66 |
| 9 | 1593 | ⟨a, b | aaa=ab, baa=b⟩ | Noc 7 | | 3 |
| 9 | 1620 | ⟨a, b | aab=aa, bbb=a⟩ | Noc Com 7 | ℕ(7 = 6) | 35 |
| 9 | 1632 | ⟨a, b | aab=ab, bbb=a⟩ | Noc Com 7 | ℕ(7 = 4) | 50 |
| 9 | 2074 | ⟨a, b | aab=b, babb=a⟩ | Noc Com 7 | | 20 |
| 9 | 2231 | ⟨a, b | ab=aa, aaa=bb⟩ | Noc 7 | | 3 |
| 9 | 2271 | ⟨a, b | bb=aa, aaa=ab⟩ | Noc 7 | | 3 |
| 9 | 3197 | ⟨a, b | ab=a, bbb=baa⟩ | Noc 7 | | 5 |
| 10 | 4162 | ⟨a, b | abb=aab, bbb=a⟩ | Noc Com 7 | ℕ(7 = 5) | 42 |
| 10 | 6251 | ⟨a, b | aaa=a, aabba=b⟩ | Noc 7 | | 1 |
| 10 | 8987 | ⟨a, b | ab=a, aaaaa=bb⟩ | Noc Com 7 | | 5 |
| 10 | 9108 | ⟨a, b | ab=a, bbbbb=aa⟩ | Noc Com 7 | | 2 |
| 10 | 9110 | ⟨a, b | ab=a, bbbbb=ba⟩ | Noc 7 | | 1 |
| 10 | 9263 | ⟨a, b | ab=a, aaaa=bbb⟩ | Noc Com 7 | | 4 |
| 10 | 9375 | ⟨a, b | ab=a, bbba=bbb⟩ | Noc 7 | | 3 |
| 10 | 9376 | ⟨a, b | ab=a, bbbb=aaa⟩ | Noc Com 7 | | 3 |
| 10 | 9382 | ⟨a, b | ab=a, bbbb=bba⟩ | Noc 7 | | 1 |
| 11 | 14843 | ⟨a, b | abba=b, aabab=a⟩ | Noc 7 | | 1 |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
5 total
| Σ | # | Presentation | Mapping |
|---|
| 11 | 20072 | ⟨a, b | aab=b, aabb=aaa⟩ | φ(a) = a, φ(b) = b |
| 11 | 20088 | ⟨a, b | aab=b, abab=aaa⟩ | φ(a) = a, φ(b) = b |
| 11 | 20120 | ⟨a, b | aab=b, baab=aaa⟩ | φ(a) = a, φ(b) = b |
| 11 | 20270 | ⟨a, b | aba=b, aabb=aaa⟩ | φ(a) = a, φ(b) = b |
| 11 | 20286 | ⟨a, b | aba=b, abba=aaa⟩ | φ(a) = a, φ(b) = b |