#995 ⟨a, b | ab=a, aaa=bb⟩
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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 ⋅ a3 = a and a ⋅ 1 = a, however a3 ≠ 1
- Commutative Gröbner basis: ⟨a, b | ab=a, a3=b2, b3=b2⟩
- Cancellative quotient is isomorphic to ℤ3
- Enveloping group is isomorphic to ℤ3
- 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 3
- b: index 2, period 1
- Histogram:
| index 1, period 1 | 1 element | a3 |
| index 1, period 3 | 2 elements | a, a2 |
| index 2, period 1 | 1 element | b |
Idempotents are shown in bold.
|
1 | a | b | a2 | a3 |
|---|
| 1 | 1 | a | b | a2 | a3 |
|---|
| a | a | a2 | a | a3 | a |
|---|
| b | b | a | a3 | a2 | a3 |
|---|
| a2 | a2 | a3 | a2 | a | a2 |
|---|
| a3 | a3 | a | a3 | a2 | a3 |
|---|
Idempotents are shown in bold.
- Reduction order:
- Left-to-right recursive path with deg(a) = 0; deg(b) = 1
- Certificate: derivations of all rewriting rules from the defining relations.
# ab:ab=a,aaa=bb a/b
aaaa=a
ba=a
ab=a
bb=aaa
11 unique, 1443 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 | 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 |
| 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.
19 total
| Σ | # | Presentation | Mapping |
|---|
| 9 | 3079 | ⟨a, b | ab=a, aaab=bb⟩ | φ(a) = a, φ(b) = b |
| 9 | 3083 | ⟨a, b | ab=a, aaba=bb⟩ | φ(a) = a, φ(b) = b |
| 9 | 3091 | ⟨a, b | ab=a, abaa=bb⟩ | φ(a) = a, φ(b) = b |
| 10 | 8999 | ⟨a, b | ab=a, aaabb=bb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9007 | ⟨a, b | ab=a, aabab=bb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9011 | ⟨a, b | ab=a, aabba=bb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9023 | ⟨a, b | ab=a, abaab=bb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9027 | ⟨a, b | ab=a, ababa=bb⟩ | φ(a) = a, φ(b) = b |
| 10 | 9035 | ⟨a, b | ab=a, abbaa=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24891 | ⟨a, b | ab=a, aaabbb=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24907 | ⟨a, b | ab=a, aababb=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24915 | ⟨a, b | ab=a, aabbab=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24919 | ⟨a, b | ab=a, aabbba=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24939 | ⟨a, b | ab=a, abaabb=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24947 | ⟨a, b | ab=a, ababab=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24951 | ⟨a, b | ab=a, ababba=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24963 | ⟨a, b | ab=a, abbaab=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24967 | ⟨a, b | ab=a, abbaba=bb⟩ | φ(a) = a, φ(b) = b |
| 11 | 24975 | ⟨a, b | ab=a, abbbaa=bb⟩ | φ(a) = a, φ(b) = b |