#280 ⟨a, b | ab=a, bb=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 7
- Finite commutative monoid with 4 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, b2=a2, a3=a⟩
- 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 2, period 1
- Histogram:
| index 1, period 1 | 1 element | a2 |
| index 1, period 2 | 1 element | a |
| index 2, period 1 | 1 element | b |
Idempotents are shown in bold.
|
1 | a | b | a2 |
|---|
| 1 | 1 | a | b | a2 |
|---|
| a | a | a2 | a | a |
|---|
| b | b | a | a2 | a2 |
|---|
| a2 | a2 | a | 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,bb=aa ab
ab=a
ba=a
bb=aa
aaa=a
8 unique, 1590 total
| Σ | # | Presentation | Description | Related |
|---|
| 5 | 7 | ⟨a, b | aa=b, bb=1⟩ | Isomorphic to ℤ4 | 1419 iso |
| 6 | 57 | ⟨a, b | aa=a, bb=a⟩ | Isomorphic to ℕ(4 = 2) | 37 iso |
| 6 | 61 | ⟨a, b | aa=b, bb=a⟩ | Isomorphic to ℕ(4 = 1) | 72 iso |
| 7 | 158 | ⟨a, b | ab=aa, ba=b⟩ | Finite non-commutative monoid with 4 elements | 8 iso, 6 anti-iso |
| 7 | 159 | ⟨a, b | ab=aa, bb=a⟩ | Isomorphic to ℕ(4 = 3) | 16 iso |
| 7 | 242 | ⟨a, b | aa=a, abb=b⟩ | Finite non-commutative monoid with 4 elements | 14 iso |
| 9 | 2881 | ⟨a, b | aa=a, abbba=b⟩ | Finite commutative monoid with 4 elements | 8 iso |
| 10 | 5033 | ⟨a, b | aaa=aa, abba=b⟩ | Finite commutative monoid with 4 elements | 2 iso |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
17 total
| Σ | # | Presentation | Mapping |
|---|
| 8 | 999 | ⟨a, b | ab=a, aab=bb⟩ | φ(a) = a, φ(b) = b |
| 8 | 1003 | ⟨a, b | ab=a, aba=bb⟩ | φ(a) = a, φ(b) = b |
| 9 | 3087 | ⟨a, b | ab=a, aabb=bb⟩ | φ(a) = a, φ(b) = b |
| 9 | 3095 | ⟨a, b | ab=a, abab=bb⟩ | φ(a) = a, φ(b) = b |
| 9 | 3099 | ⟨a, b | ab=a, abba=bb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9015 | ⟨a, b | ab=a, aabbb=bb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9031 | ⟨a, b | ab=a, ababb=bb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9039 | ⟨a, b | ab=a, abbab=bb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9043 | ⟨a, b | ab=a, abbba=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 12437 | ⟨a, b | aabb=aa, abbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12501 | ⟨a, b | abab=aa, abbb=b⟩ | φ(a) = b, φ(b) = a |
| 11 | 12594 | ⟨a, b | abba=bb, baaa=a⟩ | φ(a) = a, φ(b) = b |
| 11 | 24923 | ⟨a, b | ab=a, aabbbb=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24955 | ⟨a, b | ab=a, ababbb=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24971 | ⟨a, b | ab=a, abbabb=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24979 | ⟨a, b | ab=a, abbbab=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24983 | ⟨a, b | ab=a, abbbba=bb⟩ | φ(a) = a, φ(b) = b |