#8987 ⟨a, b | ab=a, aaaaa=bb⟩
Contents
- Properties
- Commutative structure
- Finite structure
- Staircase diagram
- Cayley table
- Right Cayley graph
- Rewriting system
- Same cardinality
- Isomorphic instances
- Sum of relation sides is 10
- Finite commutative monoid with 7 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=b2, a5=b2⟩
- Cancellative quotient is isomorphic to ℤ5
- Enveloping group is isomorphic to ℤ5
- 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 5
- b: index 2, period 1
- Histogram:
| index 1, period 1 | 1 element | b2 |
| index 1, period 5 | 4 elements | a, a2, a3, a4 |
| index 2, period 1 | 1 element | b |
Idempotents are shown in bold.
|
1 | a | b | a2 | b2 | a3 | a4 |
|---|
| 1 | 1 | a | b | a2 | b2 | a3 | a4 |
|---|
| a | a | a2 | a | a3 | a | a4 | b2 |
|---|
| b | b | a | b2 | a2 | b2 | a3 | a4 |
|---|
| a2 | a2 | a3 | a2 | a4 | a2 | b2 | a |
|---|
| b2 | b2 | a | b2 | a2 | b2 | a3 | a4 |
|---|
| a3 | a3 | a4 | a3 | b2 | a3 | a | a2 |
|---|
| a4 | a4 | b2 | a4 | a | a4 | a2 | 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,aaaaa=bb ab - -
ab=a
ba=a
bbb=bb
aaaaa=bb
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 | 1601 | ⟨a, b | aaa=bb, aab=b⟩ | Noc Com 7 | | 5 |
| 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 | 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 | 24867 | ⟨a, b | ab=a, aaaaab=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24871 | ⟨a, b | ab=a, aaaaba=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24879 | ⟨a, b | ab=a, aaabaa=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24895 | ⟨a, b | ab=a, aabaaa=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24927 | ⟨a, b | ab=a, abaaaa=bb⟩ | φ(a) = a, φ(b) = b |