#26639 ⟨a, b | aa=1, abbbbbbb=b⟩
Contents
- Properties
- Finite structure
- Right Cayley graph
- Left Cayley graph
- Rewriting system
- Same cardinality
- Isomorphic instances
- Sum of relation sides is 11
- Finite non-commutative monoid with 26 elements
- Not cancellative, because multiplication by b is not injective:
-
b ⋅ ab6 = b and b ⋅ 1 = b, however ab6 ≠ 1
Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The order of x is the least n (if it exists) such that xn = 1.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.
- 1 element center:
- 2 non-trivial idempotents:
- Order of generators:
- a: order 2
- b: index 1, period 12
- Histogram:
| order 2 | 1 element | a |
| index 1, period 1 | 2 elements | ab6, b6a |
| index 1, period 2 | 2 elements | b6, ab6a |
| index 1, period 3 | 4 elements | ab2, b2a, b4, ab4a |
| index 1, period 4 | 4 elements | b3, ab3, b3a, ab3a |
| index 1, period 6 | 4 elements | b2, ab2a, ab4, b4a |
| index 1, period 12 | 8 elements | b, ab, ba, aba, b5, ... |
- Reduction order:
- Left-to-right shortlex with a < b
- Certificate: derivations of all rewriting rules from the defining relations.
# ab:aa=1,abbbbbbb=b ab - -
aa=1
bab=abb
bbbbbbb=ab
5 unique, 25 total
| Σ | # | Presentation | Properties | φ |
|---|
| 9 | 3645 | ⟨a, b | aa=1, abab=bbb⟩ | Noc 26 | 19 |
| 10 | 5088 | ⟨a, b | aaa=bb, baab=a⟩ | Noc 26 | 1 |
| 10 | 7050 | ⟨a, b | bb=aa, ababa=a⟩ | Noc 26 | |
| 11 | 15941 | ⟨a, b | aba=bb, bbabb=a⟩ | Noc 26 | |
| 11 | 19766 | ⟨a, b | aba=b, babbb=aa⟩ | Noc 26 | |
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
1 total
| Σ | # | Presentation | Mapping |
|---|
| 11 | 27238 | ⟨a, b | aa=1, bbbbbbb=ab⟩ | φ(a) = a, φ(b) = b |