#5033 ⟨a, b | aaa=aa, abba=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 10
- Finite commutative monoid with 4 elements
- Not cancellative, because multiplication by b is not injective:
-
b ⋅ a = b and b ⋅ 1 = b, however a ≠ 1
- Gröbner basis: ⟨a, b | b2=b, ab=b, a3=a2⟩
- Cancellative quotient is isomorphic to ℤ1
- Enveloping group is isomorphic to ℤ1
- Group of units is isomorphic to ℤ1
- 3 Archimedian components:
The zero element z satisfies zy = yz = z for all y.
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.
- Zero element: b
- 2 non-trivial idempotents:
- Order of generators:
- a: index 2, period 1
- b: index 1, period 1
- Histogram:
| index 1, period 1 | 2 elements | b, a2 |
| index 2, period 1 | 1 element | a |
Idempotents are shown in bold.
|
1 | a | b | a2 |
|---|
| 1 | 1 | a | b | a2 |
|---|
| a | a | a2 | b | a2 |
|---|
| b | b | b | b | b |
|---|
| a2 | a2 | a2 | b | a2 |
|---|
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=aa,abba=b ab - -
ab=b
ba=b
bb=b
aaa=aa
8 unique, 1605 total
| Σ | # | Presentation | Properties | Description | φ |
|---|
| 5 | 7 | ⟨a, b | aa=b, bb=1⟩ | Grp Com 4 | ℤ4 | 1419 |
| 6 | 57 | ⟨a, b | aa=a, bb=a⟩ | Noc Com 4 | ℕ(4 = 2) | 37 |
| 6 | 61 | ⟨a, b | aa=b, bb=a⟩ | Noc Com 4 | ℕ(4 = 1) | 72 |
| 7 | 158 | ⟨a, b | ab=aa, ba=b⟩ | Noc 4 | | 14 |
| 7 | 159 | ⟨a, b | ab=aa, bb=a⟩ | Noc Com 4 | ℕ(4 = 3) | 16 |
| 7 | 242 | ⟨a, b | aa=a, abb=b⟩ | Noc 4 | | 14 |
| 7 | 280 | ⟨a, b | ab=a, bb=aa⟩ | Noc Com 4 | | 17 |
| 9 | 2881 | ⟨a, b | aa=a, abbba=b⟩ | Noc Com 4 | | 8 |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
2 total
| Σ | # | Presentation | Mapping |
|---|
| 11 | 15414 | ⟨a, b | aaa=aa, aabba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 15420 | ⟨a, b | aaa=aa, ababa=b⟩ | φ(a) = a, φ(b) = b |