#1020 ⟨a, b | ab=a, bbb=aa⟩
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- Properties
- Elements
- Staircase diagram
- Cayley table
- Right Cayley graph
- Rewriting system
- Same cardinality
- Isomorphic instances
- Presentation has sum-of-sides 8
- Finite commutative monoid with 5 elements
- Not cancellative, because multiplication by a is not injective:
-
a ⋅ a2 = a and a ⋅ 1 = a, however a2 ≠ 1
- Commutative Gröbner basis: ⟨a, b | ab=a, a3=a, b3=a2⟩
- Cancellative quotient is isomorphic to ℤ2
- Enveloping group is isomorphic to ℤ2
- 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 2
- b: index 3, period 1
- Histogram:
| index 1, period 1 | 1 element | a2 |
| index 1, period 2 | 1 element | a |
| index 2, period 1 | 1 element | b2 |
| index 3, period 1 | 1 element | b |
Idempotents are shown in bold.
|
1 | a | b | a2 | b2 |
|---|
| 1 | 1 | a | b | a2 | b2 |
|---|
| a | a | a2 | a | a | a |
|---|
| b | b | a | b2 | a2 | a2 |
|---|
| a2 | a2 | a | a2 | a2 | a2 |
|---|
| b2 | b2 | a | a2 | a2 | 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:ab=a,bbb=aa ab
ab=a
ba=a
aaa=a
bbb=aa
11 unique, 1453 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 | 1022 | ⟨a, b | ab=a, bbb=ba⟩ | Finite non-commutative monoid with 5 elements | 4 iso |
| 10 | 8617 | ⟨a, b | aa=a, abbbba=b⟩ | Finite commutative monoid with 5 elements | 3 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.
9 total
| Σ | # | Presentation | Mapping |
|---|
| 9 | 3194 | ⟨a, b | ab=a, bbb=aab⟩ | φ(a) = a, φ(b) = b |
| 9 | 3195 | ⟨a, b | ab=a, bbb=aba⟩ | φ(a) = a, φ(b) = b |
| 10 | 9287 | ⟨a, b | ab=a, aabb=bbb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9303 | ⟨a, b | ab=a, abab=bbb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9311 | ⟨a, b | ab=a, abba=bbb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25467 | ⟨a, b | ab=a, aabbb=bbb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25499 | ⟨a, b | ab=a, ababb=bbb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25515 | ⟨a, b | ab=a, abbab=bbb⟩ | φ(a) = a, φ(b) = b |
| 11 | 25523 | ⟨a, b | ab=a, abbba=bbb⟩ | φ(a) = a, φ(b) = b |