#4125 ⟨a, b | aab=aaa, baa=b⟩
Contents
- Properties
- Finite structure
- Cayley table
- Right Cayley graph
- Left Cayley graph
- Rewriting system
- Same cardinality
- Isomorphic instances
- Anti-isomorphic instances
- Sum of relation sides is 10
- Finite non-commutative monoid with 9 elements
- Not cancellative, because multiplication by b is not injective:
-
b ⋅ b2 = b and b ⋅ 1 = b, however b2 ≠ 1
Elements in the center commute with all other elements.
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 element center:
- 3 non-trivial idempotents:
- Order of generators:
- a: index 3, period 2
- b: index 1, period 2
- Histogram:
| index 1, period 1 | 3 elements | ab, b2, a2b2 |
| index 1, period 2 | 3 elements | b, a2b, ab2 |
| index 2, period 1 | 1 element | a2 |
| index 3, period 2 | 1 element | a |
Idempotents are shown in bold.
|
1 | a | b | a2 | ab | b2 | a2b | ab2 | a2b2 |
|---|
| 1 | 1 | a | b | a2 | ab | b2 | a2b | ab2 | a2b2 |
|---|
| a | a | a2 | ab | a2b | a2b | ab2 | a2b2 | a2b2 | a2b |
|---|
| b | b | b2 | b2 | b | b | b | b2 | b2 | b |
|---|
| a2 | a2 | a2b | a2b | a2b2 | a2b2 | a2b2 | a2b | a2b | a2b2 |
|---|
| ab | ab | ab2 | ab2 | ab | ab | ab | ab2 | ab2 | ab |
|---|
| b2 | b2 | b | b | b2 | b2 | b2 | b | b | b2 |
|---|
| a2b | a2b | a2b2 | a2b2 | a2b | a2b | a2b | a2b2 | a2b2 | a2b |
|---|
| ab2 | ab2 | ab | ab | ab2 | ab2 | ab2 | ab | ab | ab2 |
|---|
| a2b2 | a2b2 | a2b | a2b | a2b2 | a2b2 | a2b2 | a2b | a2b | a2b2 |
|---|
Idempotents are shown in bold.
Idempotents are shown in bold.
- Reduction order:
- Left-to-right shortlex with b < a
- Certificate: derivations of all rewriting rules from the defining relations.
# ab:aab=aaa,baa=b ba - -
ba=bb
bbb=b
aaa=aab
26 unique, 940 total
| Σ | # | Presentation | Properties | Description | φ |
|---|
| 7 | 118 | ⟨a, b | aaa=b, bbb=1⟩ | Grp Com 9 | ℤ9 | 562 |
| 8 | 568 | ⟨a, b | aaa=a, abb=b⟩ | Noc 9 | | 5 |
| 8 | 571 | ⟨a, b | aaa=a, bbb=a⟩ | Noc Com 9 | ℕ(9 = 3) | 48 |
| 8 | 581 | ⟨a, b | aaa=b, bbb=a⟩ | Noc Com 9 | ℕ(9 = 1) | 62 |
| 8 | 601 | ⟨a, b | aba=b, bab=a⟩ | Noc 9 | | 3 |
| 9 | 1584 | ⟨a, b | aaa=aa, bbb=a⟩ | Noc Com 9 | ℕ(9 = 6) | 16 |
| 9 | 1595 | ⟨a, b | aaa=ab, bab=b⟩ | Noc 9 | | 9 |
| 9 | 1598 | ⟨a, b | aaa=ab, bbb=a⟩ | Noc Com 9 | ℕ(9 = 4) | 40 |
| 9 | 1608 | ⟨a, b | aaa=bb, bbb=a⟩ | Noc Com 9 | ℕ(9 = 2) | 70 |
| 9 | 1991 | ⟨a, b | aaa=a, abba=b⟩ | Noc 9 | | 3 |
| 9 | 2064 | ⟨a, b | aab=b, abba=a⟩ | Noc 9 | | 38 |
| 9 | 2255 | ⟨a, b | ab=aa, bba=bb⟩ | Noc 9 | | |
| 10 | 4130 | ⟨a, b | aab=aaa, bbb=a⟩ | Noc Com 9 | ℕ(9 = 7) | 1 |
| 10 | 4156 | ⟨a, b | abb=aaa, bbb=a⟩ | Noc Com 9 | ℕ(9 = 5) | 12 |
| 10 | 5035 | ⟨a, b | aaa=aa, abbb=b⟩ | Noc 9 | | 3 |
| 10 | 5079 | ⟨a, b | aaa=bb, aaba=b⟩ | Noc Com 9 | | 8 |
| 10 | 5124 | ⟨a, b | aab=aa, bbbb=a⟩ | Noc Com 9 | ℕ(9 = 8) | 5 |
| 10 | 5339 | ⟨a, b | aaa=bb, aab=ab⟩ | Noc 9 | | 1 |
| 10 | 6824 | ⟨a, b | aab=b, bbb=aaa⟩ | Noc Com 9 | | 5 |
| 10 | 7139 | ⟨a, b | bb=aa, aaaa=ab⟩ | Noc 9 | | 3 |
| 10 | 9327 | ⟨a, b | ab=a, baaa=bbb⟩ | Noc 9 | | 3 |
| 10 | 9359 | ⟨a, b | ab=a, bbaa=bbb⟩ | Noc 9 | | 2 |
| 11 | 19506 | ⟨a, b | aab=b, aaaaa=ba⟩ | Noc 9 | | |
| 11 | 19507 | ⟨a, b | aab=b, aaaaa=bb⟩ | Noc Com 9 | | |
| 11 | 20723 | ⟨a, b | ab=aa, aaaaa=bb⟩ | Noc 9 | | 15 |
| 11 | 25656 | ⟨a, b | ab=a, bbbbb=baa⟩ | Noc 9 | | |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
1 total
| Σ | # | Presentation | Mapping |
|---|
| 10 | 4129 | ⟨a, b | aab=aaa, bba=b⟩ | φ(a) = a, φ(b) = b |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
1 total
| Σ | # | Presentation | Mapping |
|---|
| 11 | 20080 | ⟨a, b | aab=b, abaa=aaa⟩ | φ(a) = a, φ(b) = bb |