#8617 ⟨a, b | aa=a, abbbba=b⟩
Quick links
- Properties
- Elements
- Staircase diagram
- Cayley table
- Right Cayley graph
- Rewriting system
- Same cardinality
- Isomorphic instances
- Presentation has sum-of-sides 10
- Finite commutative monoid with 5 elements
- Not cancellative, because multiplication by a is not injective:
-
a ⋅ a = a and a ⋅ 1 = a, however a ≠ 1
- Commutative Gröbner basis: ⟨a, b | a2=a, ab=b, b4=b⟩
- Cancellative quotient is isomorphic to ℤ3
- Enveloping group is isomorphic to ℤ3
- 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 3
- Histogram:
| index 1, period 1 | 2 elements | a, b3 |
| index 1, period 3 | 2 elements | b, b2 |
Idempotents are shown in bold.
|
1 | a | b | b2 | b3 |
|---|
| 1 | 1 | a | b | b2 | b3 |
|---|
| a | a | a | b | b2 | b3 |
|---|
| b | b | b | b2 | b3 | b |
|---|
| b2 | b2 | b2 | b3 | b | b2 |
|---|
| b3 | b3 | b3 | b | b2 | b3 |
|---|
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,abbbba=b ab
aa=a
ab=b
ba=b
bbbb=b
11 unique, 1459 total
| Σ | # | Presentation | Description | Related |
|---|
| 6 | 44 | ⟨a, b | aa=b, abb=1⟩ | Isomorphic to ℤ5 | 1132 iso |
| 7 | 253 | ⟨a, b | aa=b, abb=a⟩ | Isomorphic to ℕ(5 = 1) | 71 iso |
| 7 | 254 | ⟨a, b | aa=b, abb=b⟩ | Isomorphic to ℕ(5 = 2) | 43 iso |
| 7 | 268 | ⟨a, b | ab=a, baa=b⟩ | Finite non-commutative monoid with 5 elements | 63 iso, 23 anti-iso |
| 8 | 574 | ⟨a, b | aaa=b, aab=b⟩ | Isomorphic to ℕ(5 = 3) | 27 iso |
| 8 | 950 | ⟨a, b | ab=a, bbbb=a⟩ | Isomorphic to ℕ(5 = 4) | 32 iso |
| 8 | 995 | ⟨a, b | ab=a, aaa=bb⟩ | Finite commutative monoid with 5 elements | 19 iso |
| 8 | 1019 | ⟨a, b | ab=a, bba=bb⟩ | Finite non-commutative monoid with 5 elements | 25 iso |
| 8 | 1020 | ⟨a, b | ab=a, bbb=aa⟩ | Finite commutative monoid with 5 elements | 9 iso |
| 8 | 1022 | ⟨a, b | ab=a, bbb=ba⟩ | Finite non-commutative monoid with 5 elements | 4 iso |
| 11 | 15426 | ⟨a, b | aaa=aa, abbba=b⟩ | Finite commutative monoid with 5 elements | |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
3 total
| Σ | # | Presentation | Mapping |
|---|
| 11 | 24101 | ⟨a, b | aa=a, aabbbba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24125 | ⟨a, b | aa=a, ababbba=b⟩ | φ(a) = a, φ(b) = b |
| 11 | 24135 | ⟨a, b | aa=a, abbabba=b⟩ | φ(a) = a, φ(b) = b |