#159 ⟨a, b | ab=aa, bb=a⟩
Contents
- Properties
- Commutative structure
- Finite structure
- Staircase diagram
- Cayley table
- Right Cayley graph
- Rewriting system
- Same cardinality
- Isomorphic instances
- Sum of relation sides is 7
- Isomorphic to ℕ(4 = 3)
- Finite commutative monoid with 4 elements
- Not cancellative, because multiplication by b3 is not injective:
-
b3 ⋅ b = b3 and b3 ⋅ 1 = b3, however b ≠ 1
- Gröbner basis: ⟨a, b | b4=b3, a=b2⟩
- Cancellative quotient is isomorphic to ℤ1
- Enveloping group is isomorphic to ℤ1
- Group of units is isomorphic to ℤ1
- 2 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: b3
- 1 non-trivial idempotent:
- Order of generators:
- a: index 2, period 1
- b: index 3, period 1
- Histogram:
| index 1, period 1 | 1 element | b3 |
| index 2, period 1 | 1 element | b2 |
| index 3, period 1 | 1 element | b |
Idempotents are shown in bold.
|
1 | b | b2 | b3 |
|---|
| 1 | 1 | b | b2 | b3 |
|---|
| b | b | b2 | b3 | b3 |
|---|
| b2 | b2 | b3 | b3 | b3 |
|---|
| b3 | b3 | b3 | b3 | b3 |
|---|
Idempotents are shown in bold.
- Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1
- Certificate: derivations of all rewriting rules from the defining relations.
# ab:ab=aa,bb=a b/a - -
bbbb=bbb
a=bb
8 unique, 1591 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 | 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 |
| 10 | 5033 | ⟨a, b | aaa=aa, abba=b⟩ | Noc Com 4 | | 2 |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
16 total
| Σ | # | Presentation | Mapping |
|---|
| 7 | 273 | ⟨a, b | ab=a, bbb=a⟩ | φ(a) = bbb, φ(b) = b |
| 8 | 974 | ⟨a, b | aa=b, aaa=bb⟩ | φ(a) = b, φ(b) = bb |
| 8 | 976 | ⟨a, b | aa=b, aab=ab⟩ | φ(a) = b, φ(b) = bb |
| 8 | 977 | ⟨a, b | aa=b, aab=ba⟩ | φ(a) = b, φ(b) = bb |
| 8 | 980 | ⟨a, b | aa=b, aba=ab⟩ | φ(a) = b, φ(b) = bb |
| 8 | 1021 | ⟨a, b | ab=a, bbb=ab⟩ | φ(a) = bbb, φ(b) = b |
| 9 | 2001 | ⟨a, b | aaa=b, aaaa=b⟩ | φ(a) = b, φ(b) = bbb |
| 9 | 3037 | ⟨a, b | aa=b, aaaa=ab⟩ | φ(a) = b, φ(b) = bb |
| 9 | 3154 | ⟨a, b | aa=b, aab=aaa⟩ | φ(a) = b, φ(b) = bb |
| 9 | 3155 | ⟨a, b | aa=b, aba=aaa⟩ | φ(a) = b, φ(b) = bb |
| 9 | 3196 | ⟨a, b | ab=a, bbb=abb⟩ | φ(a) = bbb, φ(b) = b |
| 10 | 9184 | ⟨a, b | aa=b, aaaa=aaa⟩ | φ(a) = b, φ(b) = bb |
| 10 | 9319 | ⟨a, b | ab=a, abbb=bbb⟩ | φ(a) = bbb, φ(b) = b |
| 11 | 19848 | ⟨a, b | aaa=b, aaaa=aaa⟩ | φ(a) = b, φ(b) = bbb |
| 11 | 19983 | ⟨a, b | aab=a, abbb=bbb⟩ | φ(a) = bbb, φ(b) = b |
| 11 | 25531 | ⟨a, b | ab=a, abbbb=bbb⟩ | φ(a) = bbb, φ(b) = b |