#2881 ⟨a, b | aa=a, abbba=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 4 elements
- Not cancellative, because multiplication by a is not injective:
-
a ⋅ a = a and a ⋅ 1 = a, however a ≠ 1
- Gröbner basis: ⟨a, b | a2=a, ab=b, b3=b⟩
- Cancellative quotient is isomorphic to ℤ2
- Enveloping group is isomorphic to ℤ2
- Group of units is isomorphic to ℤ1
- 3 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.
- 2 non-trivial idempotents:
- Order of generators:
- a: index 1, period 1
- b: index 1, period 2
- Histogram:
| index 1, period 1 | 2 elements | a, b2 |
| index 1, period 2 | 1 element | b |
Idempotents are shown in bold.
|
1 | a | b | b2 |
|---|
| 1 | 1 | a | b | b2 |
|---|
| a | a | a | b | b2 |
|---|
| b | b | b | b2 | b |
|---|
| b2 | b2 | b2 | b | b2 |
|---|
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:aa=a,abbba=b ab - -
aa=a
ab=b
ba=b
bbb=b
8 unique, 1599 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 |
| 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.
8 total
| Σ | # | Presentation | Mapping |
|---|
| 10 | 8595 | ⟨a, b | aa=a, aabbba=b⟩ | φ(a) = a, φ(b) = b |
| 10 | 8607 | ⟨a, b | aa=a, ababba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24073 | ⟨a, b | aa=a, aaabbba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24087 | ⟨a, b | aa=a, aababba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24093 | ⟨a, b | aa=a, aabbaba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24097 | ⟨a, b | aa=a, aabbbaa=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24113 | ⟨a, b | aa=a, abaabba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24119 | ⟨a, b | aa=a, abababa=b⟩ | φ(a) = a, φ(b) = b |